813:
52:
1351:. One reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient; this however, often still requires working with the original non-Hausdorff topological group. Other reasons, and some equivalent conditions, are discussed below.
7656:
1426:
of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphismβthe inverse must also be continuous. There are examples of topological groups that are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group
5198:
of a field are profinite groups.) Furthermore, every connected locally compact group is an inverse limit of connected Lie groups. At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group. (For example, the
6622:
5868:). By contrast, representation theory for topological groups that are not locally compact has so far been developed only in special situations, and it may not be reasonable to expect a general theory. For example, there are many abelian
8190:
7444:
8477:
6870:
8363:
8662:
9231:
11809:
11272:
7439:
889:
is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.
7254:
6199:. As a result, there is an essentially complete description of the possible homotopy types of Lie groups. For example, a compact connected Lie group of dimension at most 3 is either a torus, the group
5150:. As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general.
4982:
is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism. In other words, it will not necessarily admit an inverse in the
2671:
4980:
8543:
7909:
7828:
9960:
8083:
3108:
10995:
6448:
9134:
8026:
2161:
10146:
9287:
10235:
4026:
3783:
3693:
9395:
6367:
9358:
11330:
4711:
11205:
11173:
2859:
6961:
9904:
9686:
9533:
11021:
3835:
2537:
2198:
2484:
8804:
8372:
6814:
12183:
9747:
9041:
8261:
3184:
is necessarily a symmetric neighborhood of the identity element. Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.
2925:
11723:
10461:
10051:
9984:
9825:
9565:
9483:
9067:
8846:
8252:
7374:
7163:
7113:
7048:
6924:
2561:
11912:
10826:
10027:
2386:
11699:
6393:
6180:
6146:
6116:
6046:
6018:
5851:
5822:
5716:
5675:
5638:
5616:
5592:
5568:
5487:
5295:
5260:
5225:
5182:
4608:
4302:
4275:
4255:
4235:
4215:
3496:
2994:
2098:
2055:
2025:
1986:
1937:
1905:
1858:
1827:
1801:
1756:
1728:
1684:
1645:
1616:
1575:
1504:
1474:
497:
472:
435:
7089:
3884:
8758:
6998:
6810:
6716:
6659:
5795:
5758:
4557:
12098:
11595:
7958:
4878:
from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups.
3037:
12130:
11976:
11944:
5020:
4911:
3459:
11073:
10104:
9874:
8572:
7283:
2614:
2331:
2302:
10954:
9712:
5080:
4052:
3946:
3430:
2357:
8088:
7718:
4380:
4333:
3920:
1774:, not just continuous. Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about
1427:
is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.
7651:{\displaystyle \Delta _{X}(N)\circ \Delta _{X}(N)=\{(x,z):{\text{ there exists }}y\in X{\text{ such that }}x,z\in y+N\}=\bigcup _{y\in X}=\Delta _{X}+(N\times N).}
7338:
8220:
12023:
11832:
11755:
11750:
11618:
11497:
11432:
10896:
10873:
10754:
10664:
10504:
10389:
10346:
10281:
9798:
9656:
9460:
9310:
9157:
8940:
8725:
7932:
7764:
6900:
6779:
6416:
6278:
2882:
2797:
2714:
12063:
12043:
12000:
11883:
11863:
11673:
11653:
11563:
11524:
11474:
11454:
11406:
11383:
11356:
11292:
11225:
11137:
11117:
11093:
11041:
10918:
10846:
10794:
10774:
10731:
10711:
10689:
10641:
10621:
10597:
10573:
10553:
10525:
10481:
10434:
10414:
10366:
10323:
10303:
10255:
10188:
10168:
10073:
9924:
9845:
9775:
9633:
9613:
9589:
9503:
9437:
9417:
9087:
9005:
8981:
8960:
8917:
8897:
8877:
8702:
8682:
8563:
7848:
7698:
7678:
7305:
6756:
6736:
6683:
6442:
6304:
3855:
3584:
2965:
2945:
2774:
2754:
2734:
2691:
2585:
2434:
2406:
4339:. By local compactness, closed balls of sufficiently small radii are compact, and by normalising we can assume this holds for radius 1. Closing the open ball,
5418:
of a compact group can be decomposed as a
Hilbert-space direct sum of irreducible representations, which are all finite-dimensional; this is part of the
1275:
This definition used notation for multiplicative groups; the equivalent for additive groups would be that the following two operations are continuous:
5681:
The irreducible unitary representations of a locally compact group may be infinite-dimensional. A major goal of representation theory, related to the
2223:, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are
6260:
This article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element
2250:
with multiplicative identity, the set of invertible elements forms a topological group under multiplication. For example, the group of invertible
9162:
1378:. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.
5056:
There are several strong results on the relation between topological groups and Lie groups. First, every continuous homomorphism of Lie groups
799:
11230:
This example also shows that complete subsets (indeed, even compact subsets) of a non-Hausdorff space may fail to be closed (for example, if
7126:
The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin.
5157:(understood to be Hausdorff) is an inverse limit of compact Lie groups. (One important case is an inverse limit of finite groups, called a
6094:. So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup of
11233:
7379:
2716:
In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element. If
5693:(the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many cases such as
7168:
12309: β topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined
847:
condition for the group operations connects these two structures together and consequently they are not independent from each other.
6309:
4913:
is a morphism of topological groups (that is, a continuous homomorphism), it is not necessarily true that the induced homomorphism
4308:
and Agata
Przybyszewska in 2006, the idea of the which is as follows: One relies on the construction of a left-invariant metric,
2618:
5403:
4916:
357:
8487:
7853:
7772:
13348:
13076:
12849:
9929:
8031:
5860:
has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of
5402:
Group actions and representation theory are particularly well understood for compact groups, generalizing what happens for
3042:
307:
10959:
6242:
Information about convergence of nets and filters, such as definitions and properties, can be found in the article about
9099:
7963:
4277:
2 holds, since in particular any properly metrisable space is countable union of compact metrisable and thus separable (
2113:
13237:
13151:
13114:
5048:
The third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.
792:
302:
17:
13275:. trans. from Russian by Arlen Brown and P.S.V. Naidu (3rd ed.). New York: Gordon and Breach Science Publishers.
10109:
9238:
13314:
13280:
13184:
13050:
13014:
12989:
12956:
12880:
12289:
10201:
3957:
3698:
3608:
9363:
12330:
5910:
3249:
is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such as
9315:
5695:
11297:
2211:; this phrase is best understood informally, to include several different families of examples. For example, a
4990:
There is a version of the first isomorphism theorem for topological groups, which may be stated as follows: if
4728:
4684:
2208:
1371:
718:
11178:
11146:
10997:
is complete (since it is clearly compact and every compact set is necessarily complete). So in particular, if
5501:
The irreducible representations of all compact connected Lie groups have been classified. In particular, the
4644:. Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.
2802:
13340:
12315:
6617:{\displaystyle \Delta _{X}(N):=\{(x,y)\in X\times X:x-y\in N\}=\bigcup _{y\in X}=\Delta _{X}+(N\times \{0\})}
5725:
785:
5082:
is smooth. It follows that a topological group has a unique structure of a Lie group if one exists. Also,
5985:, this puts strong restrictions on the possible cohomology rings of topological groups. In particular, if
3531:
6933:
2200:
are given the discrete topology. Another large class of pro-finite groups important in number theory are
13376:
13176:
9879:
9661:
9508:
4394:, the subgroup has at most countably many cosets. One now uses this sequence of cosets and the metric on
402:
216:
11000:
3788:
2489:
2166:
13306:
13260:
13042:
12251:
is a semitopological group in which the function mapping elements to their inverses is also continuous.
6049:
5865:
5411:
5328:
5094:
2439:
1950:
134:
12292: β A TVS where points that get progressively closer to each other will always converge to a point
12192:
Various generalizations of topological groups can be obtained by weakening the continuity conditions:
8763:
13386:
12335:
12135:
9717:
9011:
5686:
2894:
11704:
10442:
10032:
9965:
9806:
9538:
9464:
9048:
8809:
8225:
7343:
7132:
7094:
7003:
6905:
2542:
2232:
1407:. A group homomorphism between topological groups is continuous if and only if it is continuous at
12599:
12376:
11888:
10802:
10003:
6076:
5682:
4881:
For example, a native version of the first isomorphism theorem is false for topological groups: if
3303:
2362:
2212:
1508:
1170:
To show that a topology is compatible with the group operations, it suffices to check that the map
886:
600:
334:
211:
99:
11678:
6372:
6163:
6129:
6099:
6029:
6001:
5834:
5805:
5699:
5658:
5621:
5599:
5575:
5551:
5470:
5406:. For example, every finite-dimensional (real or complex) representation of a compact group is a
5278:
5243:
5208:
5165:
4591:
4287:
4260:
4240:
4220:
4200:
3472:
2973:
2081:
2038:
2008:
1969:
1920:
1888:
1841:
1810:
1784:
1739:
1711:
1667:
1628:
1599:
1558:
1487:
1457:
480:
455:
418:
13213:
7743:
has its own definition of a "Cauchy prefilter" and "Cauchy net." For the canonical uniformity on
7053:
5083:
3860:
3250:
870:
8730:
6965:
6786:
6692:
6629:
5771:
5734:
4522:
12833:
12594:
12248:
12068:
11568:
11538:
11527:
10532:
10528:
9995:
7937:
6237:
5506:
5419:
5415:
4391:
4149:
3003:
750:
540:
12103:
11949:
11917:
8185:{\displaystyle x_{\bullet }\times y_{\bullet }:=\left(x_{i},y_{j}\right)_{(i,j)\in I\times J}}
4993:
4884:
4672:
becomes a topological group when given the quotient topology. It is
Hausdorff if and only if
3435:
1699:
can be viewed as studying the structure of the orthogonal group, or the closely related group
12904:
12367:
12327: β Topological group that is in a certain sense assembled from a system of finite groups
12306:
12301:
12255:
12197:
11046:
10080:
9850:
7259:
6222:
5517:
5355:
5195:
4336:
4166:
4152:
4086:
2590:
2307:
2278:
2228:
2201:
1996:
624:
12899:
10930:
9691:
5731:, every irreducible unitary representation has dimension 1. In this case, the unitary dual
5537:
5059:
4866:
is closed. Every discrete subgroup of a
Hausdorff commutative topological group is closed.
4031:
3925:
3409:
2336:
2163:
in such a way that its topology is induced by the product topology, where the finite groups
1590:
with real entries can be viewed as a topological group with the topology defined by viewing
13324:
13290:
13221:
13194:
13161:
13124:
13060:
13024:
12966:
12925:
12890:
12859:
7703:
6091:
5541:
5103:
4780:
4358:
4311:
4281:
4197:
As with the rest of the article we of assume here a
Hausdorff topology. The implications 4
3893:
3254:
1587:
1547:
564:
552:
170:
104:
12298: β Group that is also a differentiable manifold with group operations that are smooth
7314:
5153:
The theorem also has consequences for broader classes of topological groups. First, every
5110:
have the structure of a smooth manifold, making the group operations smooth? As shown by
1443:. In this sense, the theory of topological groups subsumes that of ordinary groups. The
8:
13381:
10257:
8472:{\displaystyle x_{\bullet }-y_{\bullet }:=\left(x_{i}-y_{j}\right)_{(i,j)\in I\times J}.}
8199:
7730:
6865:{\displaystyle \left\{\Delta (N):N{\text{ is a neighborhood of }}0{\text{ in }}X\right\}}
6243:
5086:
says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth
4875:
4791:
In any commutative topological group, the product (assuming the group is multiplicative)
3512:
is not
Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group
3258:
2236:
2064:
1621:
1444:
974:
882:
844:
836:
139:
34:
12005:
11814:
11732:
11600:
11479:
11414:
10878:
10855:
10736:
10646:
10486:
10371:
10328:
10263:
9780:
9638:
9442:
9292:
9139:
8922:
8707:
8358:{\displaystyle x_{\bullet }+y_{\bullet }:=\left(x_{i}+y_{j}\right)_{(i,j)\in I\times J}}
7914:
7746:
6882:
6761:
6398:
6263:
3110:
The closure of every symmetric set in a commutative topological group is symmetric. If
2864:
2779:
2696:
13332:
12929:
12353:
12048:
12028:
11985:
11868:
11848:
11658:
11638:
11548:
11509:
11459:
11439:
11391:
11368:
11341:
11277:
11210:
11122:
11102:
11078:
11026:
10903:
10831:
10779:
10759:
10716:
10696:
10674:
10626:
10606:
10582:
10558:
10538:
10510:
10466:
10419:
10399:
10351:
10308:
10288:
10240:
10173:
10153:
10058:
9909:
9830:
9760:
9754:
9618:
9598:
9574:
9488:
9422:
9402:
9072:
8990:
8966:
8945:
8902:
8882:
8862:
8687:
8667:
8548:
7833:
7683:
7663:
7290:
7123:
The canonical uniformity on any commutative topological group is translation-invariant.
6873:
6741:
6721:
6668:
6427:
6289:
6155:
6023:
5761:
4724:
4094:
3840:
3569:
3527:
2950:
2930:
2759:
2739:
2719:
2676:
2570:
2564:
2419:
2391:
1696:
1688:
1391:
878:
124:
96:
13036:
1435:
Every group can be trivially made into a topological group by considering it with the
13354:
13344:
13310:
13276:
13243:
13233:
13180:
13147:
13110:
13082:
13072:
13046:
13010:
12985:
12952:
12933:
12876:
12845:
12567:
12344:
11726:
11542:
11408:
8482:
7734:
6071:
is an exterior algebra on generators of odd degree. Moreover, a connected Lie group
5943:
5935:
5889:
5545:
5532:
of irreducible unitary representations. (The decomposition is essentially unique if
5513:
4512:
4508:
4415:
1436:
1419:
907:
840:
372:
266:
5880:
Topological groups are special among all topological spaces, even in terms of their
695:
13343:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
13201:
12940:
12913:
12841:
12358:
6151:
6057:
5898:
5502:
5115:
4304:
4 was first proved by
Raimond Struble in 1974. An alternative approach was made by
4063:
3555:
3501:
A subgroup of a commutative topological group is discrete if and only if it has an
3322:
2251:
2107:
1954:
1649:
1375:
988:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
11501:
This implies that every locally compact commutative topological group is complete.
3203:
of the identity element, there exists a symmetric relatively compact neighborhood
13320:
13298:
13286:
13217:
13190:
13157:
13143:
13135:
13120:
13106:
13098:
13056:
13020:
13006:
12962:
12948:
12921:
12886:
12872:
12855:
12324:
12268:
10924:
8852:
6188:
6053:
5869:
5529:
5528:. Every unitary representation of a locally compact group can be described as a
5158:
5139:
4984:
4652:
4082:
4067:
3359:
3336:
3192:
1867:
1804:
1771:
1767:
1732:
1543:
1477:
1404:
1363:
1348:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
12591:
Proper metrics on locally compact groups, and proper affine isometric actions on
8657:{\displaystyle \left(x_{i}-x_{j}\right)_{(i,j)\in I\times I}\to 0{\text{ in }}X}
1781:
An example of a topological group that is not a Lie group is the additive group
1354:
This article will not assume that topological groups are necessarily
Hausdorff.
1343:
Although not part of this definition, many authors require that the topology on
850:
Topological groups have been studied extensively in the period of 1925 to 1940.
13268:
12973:
12274:
5423:
5147:
5135:
5111:
4660:
4577:
3502:
2247:
1440:
863:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
6192:
5426:
describes the decomposition of the unitary representation of the circle group
851:
843:, i.e. they are groups and topological spaces at the same time, such that the
13370:
13358:
13247:
13232:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
13168:
13032:
12283:
11534:
7740:
7308:
6686:
6255:
6204:
5881:
5154:
5127:
4305:
3234:
2997:
2220:
2068:
1881:
1834:
1833:
space, and it does not have the discrete topology. An important example for
1692:
1512:
1447:(i.e. the trivial topology) also makes every group into a topological group.
1423:
690:
612:
446:
319:
185:
13086:
10555:
is endowed with the uniformity induced on it by the canonical uniformity of
9226:{\displaystyle {\mathcal {B}}-{\mathcal {B}}:=\{B-C:B,C\in {\mathcal {B}}\}}
1546:
are important examples of non-abelian topological groups. For instance, the
855:
13131:
13094:
11804:{\displaystyle \left\{\operatorname {cl} _{C}N:N\in {\mathcal {N}}\right\}}
6196:
5982:
5974:
5690:
5525:
4162:
2216:
1874:
1516:
1367:
910:
that is also a group such that the group operation (in this case product):
817:
545:
244:
233:
180:
155:
150:
109:
80:
43:
6249:
5130:
structure. Using the smooth structure, one can define the Lie algebra of
12998:
8369:
is defined to be the image of the product net under the subtraction map:
5087:
2240:
1775:
1525:
1451:
828:
12810:
12642:
5540:, which includes the most important examples such as abelian groups and
3241:
turns all left multiplications into uniformly continuous maps while the
13205:
12917:
12654:
12547:
12347: β Algebraic structure with addition, multiplication, and division
12258:
is a group with a topology such that the group operation is continuous.
5978:
5955:
5914:
5407:
5310:
5119:
4176:
4108:
2413:
2224:
1946:
1660:
712:
440:
1507:
is also a topological group under addition, and more generally, every
12981:
12800:
12798:
12796:
12794:
12769:
12767:
12765:
12513:
12511:
12509:
12507:
12295:
10368:
is not
Hausdorff then a prefilter may converge to multiple points in
5989:
is a path-connected topological group whose rational cohomology ring
1830:
1763:
533:
12505:
12503:
12501:
12499:
12497:
12495:
12493:
12491:
12489:
12487:
11267:{\displaystyle \varnothing \neq S\subseteq \operatorname {cl} \{0\}}
6022:
is finite-dimensional in each degree, then this ring must be a free
4414:
of a topological group is itself a topological group when given the
4137:
Furthermore, the following are equivalent for any topological group
4070:) states that the following three conditions on a topological group
2304:
left or right multiplication by this element yields a homeomorphism
1945:
is well behaved in that it is compact (in fact, homeomorphic to the
12678:
12666:
12630:
12606:
7434:{\displaystyle \Delta _{X}(N)^{\operatorname {op} }=\Delta _{X}(N)}
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as a direct integral of the irreducible unitary representations of
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be a topological group. As with any topological space, we say that
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turns all right multiplications into uniformly continuous maps. If
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12318: β relatively new C*-algebraic approach toward quantum groups
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In short, there is no requirement that these Cauchy prefilters on
4731:
containing the identity element) is a closed normal subgroup. If
1476:
with the usual topology form a topological group under addition.
12484:
12277: β algebraic structure that is complete relative to a metric
5928:
5490:*). Each of these representations occurs with multiplicity 1 in
5022:
is a continuous homomorphism, then the induced homomorphism from
1403:. Topological groups, together with their homomorphisms, form a
412:
326:
12523:
7249:{\displaystyle \Delta _{X}:=\Delta _{X}(\{0\})=\{(x,x):x\in X\}}
6783:
That is, it is the upward closure of the following prefilter on
1374:, in the same way that ordinary groups are group objects in the
12779:
12726:
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4344:
3158:
of the identity element, there exists a symmetric neighborhood
51:
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10623:(or equivalently, every elementary Cauchy filter/prefilter on
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It suffices to check any of the above condition for any given
7119:
if it has a base of entourages that is translation-invariant.
6187:
Finally, compact connected Lie groups have been classified by
1511:
forms an (abelian) topological group. Some other examples of
866:
are special cases of a very wide class of topological groups.
27:
Group that is a topological space with continuous group action
12690:
10194:
if it satisfies any of the following equivalent conditions:
9093:
if it satisfies any of the following equivalent conditions:
7680:
by the canonical uniformity is the same as the topology that
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6200:
5927:. Some of these restrictions hold in the broader context of
5512:
More generally, locally compact groups have a rich theory of
4757:. So the collection of all left cosets (or right cosets) of
4495:
5923:; that implies various restrictions on the homotopy type of
5905:
over topological spaces, under mild hypotheses). The group
5872:
for which every representation on
Hilbert space is trivial.
5760:
is a group, in fact another locally compact abelian group.
10531:
uniform space (under the point-set topology definition of "
3302:
As a uniform space, every commutative topological group is
2666:{\displaystyle x{\mathcal {N}}:=\{xN:N\in {\mathcal {N}}\}}
1103:}. The inversion map is continuous if and only if for any
12832:
12816:
10875:
The same can be said of the convergence of Cauchy nets in
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will be complete even if some or all Cauchy prefilters on
6082:, which is unique up to conjugation, and the inclusion of
9989:
4975:{\displaystyle {\tilde {f}}:G/\ker f\to \mathrm {Im} (f)}
8538:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
7904:{\displaystyle y_{\bullet }=\left(y_{j}\right)_{j\in J}}
7823:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
5875:
3310:
with identity element 1, the following are equivalent:
3306:. Consequently, for a multiplicative topological group
12444:
12349:
Pages displaying short descriptions of redirect targets
6250:
Canonical uniformity on a commutative topological group
1949:), but it differs from (real) Lie groups in that it is
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9955:{\displaystyle x\in \operatorname {cl} {\mathcal {B}}}
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these reduces down to the definition described below.
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is an open subset of a commutative topological group
3103:{\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.}
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The product map is continuous if and only if for any
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Pages displaying wikidata descriptions as a fallback
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11362:
if any of the following equivalent conditions hold:
10990:{\displaystyle S\subseteq \operatorname {cl} \{0\},}
5548:, which decomposes the action of the additive group
5466:
is viewed as a subgroup of the multiplicative group
5311:
Representations of compact or locally compact groups
2436:
if and only if this is true of its left translation
1205:
is continuous. Explicitly, this means that for any
11655:is a completion of a commutative topological group
11629:if it is a sequentially complete subset of itself.
5888:determines a path-connected topological space, the
5505:of each irreducible representation is given by the
4089:(equivalently: the identity element 1 is closed in
3530:of the identity. This is equivalent to taking the
3399:is a neighborhood basis of the identity element in
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9129:{\displaystyle {\mathcal {B}}-{\mathcal {B}}\to 0}
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12361: β ring where ring operations are continuous
4840:is a subgroup of a commutative topological group
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4257:1 hold in any topological space. In particular 3
3114:is any subset of a commutative topological group
1366:, topological groups can be defined concisely as
13368:
13200:
13005:. Graduate Texts in Mathematics (1st ed.).
12660:
12648:
12553:
12338: β topological group whose group is abelian
12271: β Algebraic variety with a group structure
10141:{\displaystyle {\mathcal {C}}\subseteq \wp (S).}
9282:{\displaystyle \{B-B:B\in {\mathcal {B}}\}\to 0}
5764:states that for a locally compact abelian group
4765:is equal to the collection of all components of
4430:is the open set given by the union of open sets
2110:; it is isomorphic to a subgroup of the product
858:(respectively in 1933 and 1940) showed that the
12286: β Topological group with compact topology
10230:{\displaystyle {\mathcal {C}}\subseteq \wp (S)}
6689:induced by the set of all canonical entourages
5884:. One basic point is that a topological group
5853:of real numbers is isomorphic to its own dual.
5801:. For example, the dual group of the integers
5122:, the answer to this problem is yes. In fact,
4805:is a closed set. Furthermore, for any subsets
4021:{\displaystyle B(r)=\{g\in G\mid d(g,1)<r\}}
3778:{\displaystyle d(x_{1}a,x_{2}a)=d(x_{1},x_{2})}
3688:{\displaystyle d(ax_{1},ax_{2})=d(x_{1},x_{2})}
2567:of the identity element in a topological group
13228:Narici, Lawrence; Beckenstein, Edward (2011).
12900:"On the existence of exotic BanachβLie groups"
12379: β Vector space with a notion of nearness
9390:{\displaystyle {\mathcal {B}}-{\mathcal {B}}.}
6872:where this prefilter forms what is known as a
5618:. The irreducible unitary representations of
5363:on a real or complex topological vector space
3118:, then the following sets are also symmetric:
13331:
12756:
12589:Haagerup, Uffe; Przybyszewska, Agata (2006),
7724:
6362:{\displaystyle \Delta _{X}:=\{(x,x):x\in X\}}
6231:
3195:commutative group, then for any neighborhood
877:, which have many applications, for example,
793:
13069:Functional Analysis: Theory and Applications
11506:When endowed with its canonical uniformity,
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4343:, of radius 1 under multiplication yields a
4182:There is a left-invariant, proper metric on
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2660:
2635:
2523:
2502:
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2452:
2275:, which by definition means that if for any
1770:in such a way that the group operations are
13173:The Theory of Unitary Group Representations
12370: β semigroup with continuous operation
9353:{\displaystyle \{B-B:B\in {\mathcal {B}}\}}
8664:or equivalently, if for every neighborhood
5106:must be a Lie group. In other words, does
3233:Every topological group can be viewed as a
13267:
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12897:
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11325:{\displaystyle S=\operatorname {cl} \{0\}}
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987:is viewed as a topological space with the
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12838:Topological Groups and Related Structures
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11811:is a neighborhood basis at the origin in
6168:
6158:, the inclusion of the circle group into
6134:
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6063:In particular, for a connected Lie group
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5516:, because they admit a natural notion of
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5041:is an isomorphism if and only if the map
4706:{\displaystyle \mathbb {R} /\mathbb {Z} }
4699:
4689:
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4284:) subsets. The non-trivial implication 1
2262:
2207:Some topological groups can be viewed as
2171:
2134:
2086:
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2013:
1974:
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1691:of all vectors. The orthogonal group is
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1390:of topological groups means a continuous
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11200:{\displaystyle \operatorname {cl} \{0\}}
11168:{\displaystyle \operatorname {cl} \{0\}}
10799:converge to points(s) in the complement
3264:
2854:{\displaystyle SU:=\{su:s\in S,u\in U\}}
1953:. More generally, there is a theory of
811:
13066:
13031:
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12399:Continuous means that for any open set
9505:-small set (that is, there exists some
4869:
4563:. For example, for a positive integer
3857:is left-invariant just in case the map
14:
13369:
13167:
12997:
12978:A Course in Abstract Harmonic Analysis
12696:
12478:
10927:, then every subset of the closure of
10325:will converge to at most one point of
9990:Complete commutative topological group
5440:. The irreducible representations of
4769:. It follows that the quotient group
2267:Every topological group's topology is
358:Classification of finite simple groups
10643:) converges to at least one point of
10305:is Hausdorff then every prefilter on
6056:on generators of even degree with an
5876:Homotopy theory of topological groups
5339:such that the corresponding function
4125:There is a right-invariant metric on
3566:, which induces the same topology on
3558:if and only if there exists a metric
8856:is a Cauchy net that is a sequence.
6956:{\displaystyle B\in {\mathcal {B}},}
4114:There is a left-invariant metric on
2254:on a Hilbert space arises this way.
1762:The groups mentioned so far are all
993:compatible with the group operations
893:
12568:"Metrics in locally compact groups"
12204:with a topology such that for each
10483:converges to at least one point of
10416:converges to at least one point of
9899:{\displaystyle {\mathcal {B}}\to x}
9681:{\displaystyle B\in {\mathcal {B}}}
9528:{\displaystyle B\in {\mathcal {B}}}
6902:a fundamental system of entourages
5640:are all 1-dimensional, of the form
5446:are all 1-dimensional, of the form
5234:contains the compact open subgroup
4680:. For example, the quotient group
4282:properties of compact metric spaces
4186:that induces the given topology on
4129:that induces the given topology on
4118:that induces the given topology on
3285:, then there exists a neighborhood
3154:in a commutative topological group
1960:, including compact groups such as
1807:, with the topology inherited from
24:
12817:Arhangel'skii & Tkachenko 2008
12187:
11791:
11710:
11476:that is also a complete subset of
11385:is complete as a subset of itself.
11016:{\displaystyle S\neq \varnothing }
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6067:, the rational cohomology ring of
5942:is abelian. (More generally, the
5098:asked whether a topological group
4959:
4956:
4852:of the identity element such that
4714:is isomorphic to the circle group
4462:is also a subgroup. Likewise, if
3830:{\displaystyle a,x_{1},x_{2}\in G}
3469:of the identity element such that
3289:of the identity element such that
3207:of the identity element such that
3162:of the identity element such that
2655:
2627:
2548:
2532:{\displaystyle Sa:=\{sa:s\in S\}.}
2193:{\displaystyle \mathbb {Z} /p^{n}}
1648:. Another classical group is the
25:
13398:
13210:Topological Transformation Groups
12290:Complete topological vector space
11010:
10809:
9069:on an additive topological group
8879:is a subset of an additive group
8545:in an additive topological group
7960:into a directed set by declaring
6879:For a commutative additive group
6738:ranges over all neighborhoods of
6154:. Since the hyperbolic plane is
2479:{\displaystyle aS:=\{as:s\in S\}}
991:. Such a topology is said to be
13071:. New York: Dover Publications.
12331:Ordered topological vector space
11338:A commutative topological group
8799:{\displaystyle x_{i}-x_{j}\in N}
6928:translation-invariant uniformity
6843: is a neighborhood of
4398:to construct a proper metric on
3541:
3228:
1695:as a topological space. Much of
1381:
50:
12582:
12178:{\displaystyle f(y)-f(x)\in V.}
11436:There exists a neighborhood of
11119:(and every Cauchy prefilter on
9742:{\displaystyle B\subseteq x+N.}
9036:{\displaystyle B-B\subseteq N.}
5950:is zero.) Also, for any field
4576:is a homogeneous space for the
3954:if and only if all open balls,
2920:{\displaystyle g\mapsto g^{-1}}
2209:infinite dimensional Lie groups
873:, are used to study continuous
869:Topological groups, along with
820:form a topological group under
13105:, vol. 1 (2nd ed.),
12945:General Topology. Chapters 1β4
12559:
12468:
12390:
12163:
12157:
12148:
12142:
11962:
11930:
11718:{\displaystyle {\mathcal {N}}}
11625:A topological group is called
11584:
11572:
11537:, a uniform space is called a
11095:would be complete even though
10456:{\displaystyle {\mathcal {C}}}
10224:
10218:
10132:
10126:
10093:
10087:
10046:{\displaystyle {\mathcal {C}}}
9979:{\displaystyle {\mathcal {B}}}
9890:
9820:{\displaystyle {\mathcal {B}}}
9560:{\displaystyle B-B\subseteq N}
9478:{\displaystyle {\mathcal {B}}}
9273:
9120:
9062:{\displaystyle {\mathcal {B}}}
8841:{\displaystyle i,j\geq i_{0}.}
8640:
8623:
8611:
8449:
8437:
8338:
8326:
8247:{\displaystyle X\times X\to X}
8238:
8165:
8153:
7979:
7967:
7642:
7630:
7611:
7608:
7596:
7590:
7578:
7575:
7507:
7495:
7486:
7480:
7464:
7458:
7428:
7422:
7400:
7393:
7369:{\displaystyle \Delta _{X}(N)}
7363:
7357:
7228:
7216:
7207:
7195:
7158:{\displaystyle \Delta _{X}(N)}
7152:
7146:
7108:{\displaystyle {\mathcal {B}}}
7043:{\displaystyle (x+z,y+z)\in B}
7031:
7007:
6981:
6969:
6919:{\displaystyle {\mathcal {B}}}
6876:of the canonical uniformity.
6832:
6826:
6705:
6699:
6645:
6633:
6611:
6593:
6574:
6559:
6547:
6544:
6489:
6477:
6468:
6462:
6341:
6329:
5781:
5744:
5422:. For example, the theory of
5066:
5006:
4969:
4963:
4952:
4926:
4897:
4735:is the identity component and
4723:In any topological group, the
4535:
4482:Quotients and normal subgroups
4291:
4264:
4244:
4224:
4204:
4006:
3994:
3970:
3964:
3909:
3897:
3867:
3772:
3746:
3737:
3705:
3682:
3656:
3647:
3615:
2901:
2556:{\displaystyle {\mathcal {N}}}
2314:
1372:category of topological spaces
1127:, there exists a neighborhood
719:Infinite dimensional Lie group
13:
1:
12898:Banaszczyk, Wojciech (1983),
12836:; Tkachenko, Mikhail (2008).
12825:
12805:Narici & Beckenstein 2011
12786:Narici & Beckenstein 2011
12774:Narici & Beckenstein 2011
12518:Narici & Beckenstein 2011
12316:Locally compact quantum group
11907:{\displaystyle D\subseteq X,}
10821:{\displaystyle X\setminus S.}
10022:{\displaystyle S\subseteq X,}
9360:is a prefilter equivalent to
7700:started with (that is, it is
6060:on generators of odd degree.
5726:locally compact abelian group
5432:on the complex Hilbert space
2381:{\displaystyle S\subseteq G,}
2257:
1916:goes to infinity. The group
13335:; Wolff, Manfred P. (1999).
12661:Montgomery & Zippin 1955
12649:Montgomery & Zippin 1955
12566:Struble, Raimond A. (1974).
12554:Montgomery & Zippin 1955
12438:
12025:there exists a neighborhood
11694:{\displaystyle X\subseteq C}
10603:if every Cauchy sequence in
10601:sequentially complete subset
6424:canonical vicinities around
6388:{\displaystyle N\subseteq X}
6175:{\displaystyle \mathbb {R} }
6141:{\displaystyle \mathbb {R} }
6124:, and the homogeneous space
6111:{\displaystyle \mathbb {R} }
6041:{\displaystyle \mathbb {Q} }
6013:{\displaystyle \mathbb {Q} }
5856:Every locally compact group
5846:{\displaystyle \mathbb {R} }
5817:{\displaystyle \mathbb {Z} }
5711:{\displaystyle \mathbb {R} }
5670:{\displaystyle \mathbb {R} }
5633:{\displaystyle \mathbb {R} }
5611:{\displaystyle \mathbb {R} }
5587:{\displaystyle \mathbb {R} }
5563:{\displaystyle \mathbb {R} }
5482:{\displaystyle \mathbb {C} }
5353:is continuous. Likewise, a
5290:{\displaystyle \mathbb {Z} }
5255:{\displaystyle \mathbb {Z} }
5220:{\displaystyle \mathbb {Q} }
5177:{\displaystyle \mathbb {Z} }
4636:is Hausdorff if and only if
4603:{\displaystyle \mathbb {R} }
4405:
4297:{\displaystyle \Rightarrow }
4270:{\displaystyle \Rightarrow }
4250:{\displaystyle \Rightarrow }
4230:{\displaystyle \Rightarrow }
4210:{\displaystyle \Rightarrow }
4062:(named after mathematicians
3491:{\displaystyle x\not \in U.}
3461:there exists a neighborhood
2989:{\displaystyle S\subseteq G}
2093:{\displaystyle \mathbb {Z} }
2050:{\displaystyle \mathbb {Q} }
2020:{\displaystyle \mathbb {Q} }
1981:{\displaystyle \mathbb {Z} }
1932:{\displaystyle \mathbb {Z} }
1900:{\displaystyle \mathbb {Z} }
1853:{\displaystyle \mathbb {Z} }
1822:{\displaystyle \mathbb {R} }
1796:{\displaystyle \mathbb {Q} }
1751:{\displaystyle \mathbb {R} }
1723:{\displaystyle \mathbb {R} }
1687:to itself that preserve the
1679:{\displaystyle \mathbb {R} }
1640:{\displaystyle \mathbb {R} }
1611:{\displaystyle \mathbb {R} }
1570:{\displaystyle \mathbb {R} }
1499:{\displaystyle \mathbb {R} }
1469:{\displaystyle \mathbb {R} }
1233:, there exist neighborhoods
1035:, there exist neighborhoods
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
13177:University of Chicago Press
13067:Edwards, Robert E. (1995).
12867:Armstrong, Mark A. (1997).
12262:
11043:a is singleton set such as
10733:converges to some point of
10077:is necessarily a subset of
7376:is symmetric (meaning that
7084:{\displaystyle x,y,z\in X.}
6184:is a homotopy equivalence.
6150:can be identified with the
5544:.) A basic example is the
5414:. An infinite-dimensional
5412:irreducible representations
4093:, and there is a countable
3879:{\displaystyle x\mapsto ax}
2673:is a neighborhood basis of
1515:topological groups are the
1430:
1418:of topological groups is a
217:List of group theory topics
10:
13403:
13307:Cambridge University Press
13261:Princeton University Press
13140:Abstract Harmonic Analysis
13103:Abstract Harmonic Analysis
13043:Cambridge University Press
11982:if for every neighborhood
10391:The same is true for nets.
9993:
9591:is commutative then also:
8753:{\displaystyle i_{0}\in I}
7728:
7725:Cauchy prefilters and nets
6993:{\displaystyle (x,y)\in B}
6805:{\displaystyle X\times X,}
6711:{\displaystyle \Delta (N)}
6654:{\displaystyle (X,\tau ),}
6253:
6235:
6232:Complete topological group
5946:on the homotopy groups of
5790:{\displaystyle {\hat {G}}}
5753:{\displaystyle {\hat {G}}}
5687:admissible representations
5367:is a continuous action of
4552:{\displaystyle q:G\to G/H}
4426:, since the complement of
13337:Topological Vector Spaces
13255:Pontrjagin, Leon (1946).
13230:Topological Vector Spaces
12757:Schaefer & Wolff 1999
12336:Topological abelian group
12093:{\displaystyle x,y\in D,}
11752:Then the family of sets
11590:{\displaystyle (X,\tau )}
11533:In the general theory of
11411:to at least one point of
11294:is closed if and only if
11175:(include those points in
10260:to at least one point of
7953:{\displaystyle I\times J}
6213:), or its quotient group
5161:. For example, the group
4179:(as a topological space).
4111:(as a topological space).
4060:BirkhoffβKakutani theorem
3032:{\displaystyle S^{-1}=S,}
1439:; such groups are called
12834:Arhangel'skii, Alexander
12383:
12377:Topological vector space
12125:{\displaystyle y-x\in U}
11971:{\displaystyle f:D\to Y}
11939:{\displaystyle f:D\to Y}
7660:The topology induced on
7515: there exists
6626:For a topological group
6077:maximal compact subgroup
5683:Langlands classification
5045:is open onto its image.
5015:{\displaystyle f:G\to H}
4987:of topological groups.
4906:{\displaystyle f:G\to H}
4466:is a normal subgroup of
3454:{\displaystyle x\neq 1,}
2947:is a homeomorphism from
2891:The inversion operation
2213:topological vector space
1766:, meaning that they are
1509:topological vector space
887:topological vector space
871:continuous group actions
335:Elementary abelian group
212:Glossary of group theory
13214:Interscience Publishers
11885:be topological groups,
11068:{\displaystyle S=\{0\}}
10671:convergence outside of
10198:Every Cauchy prefilter
10170:of a topological group
10099:{\displaystyle \wp (S)}
9869:{\displaystyle x\in X.}
9595:For every neighborhood
9593:
9399:For every neighborhood
7278:{\displaystyle 0\in N.}
5973:has the structure of a
5938:of a topological group
5359:of a topological group
5323:on a topological space
5319:of a topological group
5194:-adic integers and the
5095:Hilbert's fifth problem
5052:Hilbert's fifth problem
4787:Closure and compactness
4626:. A homogeneous space
4418:. Every open subgroup
3281:contains a compact set
3261:on topological groups.
3225:is symmetric as well).
2927:on a topological group
2887:Symmetric neighborhoods
2609:{\displaystyle x\in X,}
2326:{\displaystyle G\to G.}
2297:{\displaystyle a\in G,}
2063:is the locally compact
1535:for any natural number
946:and the inversion map:
835:are the combination of
12685:Hewitt & Ross 1970
12572:Compositio Mathematica
12463:Hewitt & Ross 1979
12420:is open in the domain
12249:quasitopological group
12179:
12126:
12094:
12059:
12039:
12019:
11996:
11972:
11940:
11908:
11879:
11859:
11828:
11805:
11746:
11719:
11695:
11669:
11649:
11614:
11591:
11559:
11539:complete uniform space
11528:complete uniform space
11520:
11493:
11470:
11450:
11428:
11402:
11379:
11352:
11326:
11288:
11268:
11221:
11201:
11169:
11133:
11113:
11089:
11069:
11037:
11017:
10991:
10950:
10949:{\displaystyle \{0\},}
10914:
10892:
10869:
10842:
10822:
10790:
10770:
10750:
10727:
10707:
10685:
10660:
10637:
10617:
10593:
10569:
10549:
10533:complete uniform space
10521:
10500:
10477:
10457:
10430:
10410:
10385:
10362:
10342:
10319:
10299:
10277:
10251:
10231:
10184:
10164:
10142:
10100:
10069:
10047:
10023:
9996:Complete uniform space
9980:
9956:
9920:
9900:
9870:
9841:
9821:
9794:
9771:
9743:
9708:
9707:{\displaystyle x\in X}
9682:
9652:
9629:
9609:
9585:
9561:
9529:
9499:
9479:
9456:
9433:
9413:
9391:
9354:
9306:
9283:
9227:
9153:
9130:
9083:
9063:
9037:
9001:
8977:
8956:
8936:
8913:
8893:
8873:
8842:
8800:
8754:
8721:
8698:
8678:
8658:
8559:
8539:
8473:
8359:
8248:
8216:
8186:
8079:
8022:
7954:
7928:
7905:
7844:
7824:
7760:
7739:The general theory of
7714:
7694:
7674:
7652:
7435:
7370:
7334:
7301:
7279:
7250:
7165:contains the diagonal
7159:
7109:
7085:
7044:
6994:
6957:
6920:
6896:
6866:
6806:
6775:
6752:
6732:
6712:
6679:
6655:
6618:
6438:
6412:
6389:
6363:
6300:
6274:
6238:Complete uniform space
6176:
6142:
6112:
6042:
6014:
5866:GelfandβRaikov theorem
5847:
5818:
5797:is the original group
5791:
5754:
5712:
5671:
5634:
5612:
5588:
5564:
5507:Weyl character formula
5483:
5416:unitary representation
5291:
5256:
5221:
5199:locally compact group
5178:
5076:
5075:{\displaystyle G\to H}
5016:
4976:
4907:
4743:, then the left coset
4707:
4604:
4553:
4376:
4355:, on which the metric
4337:first countable spaces
4329:
4298:
4271:
4251:
4231:
4211:
4095:basis of neighborhoods
4048:
4047:{\displaystyle r>0}
4022:
3942:
3941:{\displaystyle a\in G}
3916:
3880:
3851:
3831:
3779:
3689:
3580:
3492:
3455:
3426:
3425:{\displaystyle x\in G}
3104:
3033:
2990:
2961:
2941:
2921:
2878:
2855:
2793:
2770:
2750:
2730:
2710:
2687:
2667:
2610:
2581:
2557:
2533:
2486:and right translation
2480:
2430:
2402:
2382:
2353:
2352:{\displaystyle a\in G}
2333:Consequently, for any
2327:
2298:
2263:Translation invariance
2202:absolute Galois groups
2194:
2157:
2094:
2051:
2021:
1997:locally compact groups
1982:
1933:
1901:
1854:
1823:
1797:
1752:
1724:
1680:
1641:
1612:
1571:
1500:
1470:
824:
751:Linear algebraic group
493:
468:
431:
13003:Topology and Geometry
12905:Mathematische Annalen
12368:Topological semigroup
12307:Locally compact group
12302:Locally compact field
12256:paratopological group
12198:semitopological group
12180:
12127:
12095:
12060:
12040:
12020:
11997:
11973:
11941:
11909:
11880:
11860:
11829:
11806:
11747:
11720:
11696:
11670:
11650:
11627:sequentially complete
11615:
11592:
11560:
11521:
11494:
11471:
11451:
11429:
11403:
11380:
11353:
11327:
11289:
11269:
11222:
11202:
11170:
11134:
11114:
11090:
11070:
11038:
11018:
10992:
10951:
10915:
10893:
10870:
10843:
10823:
10791:
10771:
10751:
10728:
10708:
10686:
10661:
10638:
10618:
10594:
10570:
10550:
10522:
10501:
10478:
10458:
10431:
10411:
10386:
10363:
10343:
10320:
10300:
10278:
10252:
10232:
10185:
10165:
10143:
10101:
10070:
10048:
10024:
9981:
9957:
9921:
9901:
9871:
9842:
9822:
9795:
9772:
9744:
9709:
9683:
9653:
9630:
9610:
9586:
9562:
9530:
9500:
9480:
9457:
9434:
9414:
9392:
9355:
9307:
9284:
9228:
9154:
9131:
9084:
9064:
9038:
9002:
8978:
8957:
8937:
8914:
8894:
8874:
8843:
8801:
8755:
8722:
8699:
8679:
8659:
8560:
8540:
8474:
8360:
8249:
8217:
8187:
8080:
8023:
7955:
7929:
7906:
7845:
7825:
7761:
7715:
7713:{\displaystyle \tau }
7695:
7675:
7653:
7529: such that
7436:
7371:
7335:
7302:
7280:
7251:
7160:
7117:translation-invariant
7110:
7086:
7045:
6995:
6958:
6921:
6897:
6867:
6807:
6776:
6753:
6733:
6713:
6680:
6656:
6619:
6439:
6413:
6390:
6364:
6301:
6275:
6177:
6143:
6113:
6043:
6015:
5909:is isomorphic in the
5848:
5819:
5792:
5755:
5713:
5672:
5635:
5613:
5589:
5570:on the Hilbert space
5565:
5542:semisimple Lie groups
5484:
5399:to itself is linear.
5307:' goes to infinity.)
5292:
5257:
5222:
5196:absolute Galois group
5179:
5077:
5017:
4977:
4908:
4848:is a neighborhood in
4708:
4605:
4554:
4377:
4375:{\displaystyle d_{0}}
4330:
4328:{\displaystyle d_{0}}
4299:
4272:
4252:
4232:
4212:
4049:
4023:
3943:
3917:
3915:{\displaystyle (G,d)}
3881:
3852:
3832:
3780:
3690:
3581:
3493:
3456:
3427:
3265:Separation properties
3150:For any neighborhood
3105:
3034:
2991:
2962:
2942:
2922:
2879:
2861:is an open subset of
2856:
2794:
2776:is an open subset of
2771:
2751:
2731:
2711:
2688:
2668:
2611:
2582:
2558:
2534:
2481:
2431:
2403:
2383:
2354:
2328:
2299:
2271:translation invariant
2233:Diffeomorphism groups
2195:
2158:
2095:
2052:
2022:
1983:
1934:
1902:
1884:of the finite groups
1855:
1824:
1798:
1753:
1725:
1681:
1642:
1613:
1572:
1501:
1471:
1219:and any neighborhood
1113:and any neighborhood
1021:and any neighborhood
815:
494:
469:
432:
13303:Topological Geometry
13212:, New York, London:
12136:
12104:
12069:
12049:
12029:
12006:
11986:
11980:uniformly continuous
11950:
11918:
11889:
11869:
11849:
11815:
11756:
11733:
11705:
11679:
11659:
11639:
11601:
11569:
11549:
11510:
11480:
11460:
11440:
11415:
11392:
11388:Every Cauchy net in
11369:
11342:
11298:
11278:
11234:
11211:
11179:
11147:
11123:
11103:
11079:
11047:
11027:
11001:
10960:
10931:
10904:
10879:
10856:
10832:
10803:
10780:
10760:
10737:
10717:
10697:
10675:
10647:
10627:
10607:
10583:
10559:
10539:
10511:
10487:
10467:
10443:
10439:Every Cauchy filter
10420:
10400:
10396:Every Cauchy net in
10372:
10352:
10329:
10309:
10289:
10264:
10241:
10202:
10174:
10154:
10110:
10081:
10059:
10033:
10004:
10000:Recall that for any
9966:
9930:
9910:
9880:
9851:
9831:
9807:
9781:
9761:
9718:
9692:
9662:
9639:
9619:
9599:
9575:
9539:
9509:
9489:
9465:
9443:
9423:
9403:
9364:
9316:
9293:
9239:
9163:
9140:
9100:
9073:
9049:
9012:
8991:
8967:
8946:
8923:
8919:is a set containing
8903:
8883:
8863:
8810:
8764:
8731:
8708:
8688:
8668:
8573:
8549:
8488:
8373:
8365:and similarly their
8262:
8226:
8200:
8089:
8032:
7964:
7938:
7915:
7854:
7834:
7773:
7747:
7704:
7684:
7664:
7445:
7380:
7344:
7333:{\displaystyle -N=N}
7315:
7291:
7260:
7169:
7133:
7095:
7054:
7004:
6966:
6934:
6906:
6883:
6815:
6787:
6762:
6742:
6722:
6693:
6669:
6663:canonical uniformity
6630:
6449:
6428:
6399:
6373:
6310:
6290:
6264:
6164:
6130:
6120:is the circle group
6100:
6092:homotopy equivalence
6030:
6002:
5835:
5825:is the circle group
5806:
5772:
5735:
5700:
5659:
5622:
5600:
5576:
5552:
5471:
5279:
5244:
5209:
5166:
5104:topological manifold
5060:
4994:
4917:
4885:
4876:isomorphism theorems
4870:Isomorphism theorems
4781:totally disconnected
4749:is the component of
4685:
4592:
4523:
4519:. The quotient map
4458:then the closure of
4359:
4335:, as in the case of
4312:
4288:
4261:
4241:
4221:
4201:
4032:
3958:
3926:
3894:
3861:
3841:
3789:
3699:
3609:
3570:
3473:
3436:
3410:
3043:
3004:
2974:
2951:
2931:
2895:
2865:
2803:
2780:
2760:
2740:
2720:
2697:
2677:
2619:
2591:
2571:
2543:
2490:
2440:
2420:
2392:
2363:
2337:
2308:
2279:
2237:homeomorphism groups
2167:
2114:
2082:
2039:
2009:
1970:
1951:totally disconnected
1921:
1889:
1842:
1811:
1785:
1740:
1712:
1668:
1629:
1600:
1559:
1548:general linear group
1488:
1458:
481:
456:
419:
13333:Schaefer, Helmut H.
8258:of these two nets:
8215:{\displaystyle X=Y}
7731:Filters in topology
6420:canonical entourage
6244:filters in topology
5375:such that for each
4729:connected component
3922:to itself for each
3532:Kolmogorov quotient
3259:uniform convergence
1659:, the group of all
1624:of Euclidean space
1445:indiscrete topology
1362:In the language of
1003:Checking continuity
883:functional analysis
125:Group homomorphisms
35:Algebraic structure
13377:Topological groups
13273:Topological Groups
13269:Pontryagin, Lev S.
13257:Topological Groups
13038:Algebraic Topology
12974:Folland, Gerald B.
12918:10.1007/BF01456956
12675:, section III.4.6.
12639:, section III.2.8.
12615:, section III.2.5.
12544:, section III.2.7.
12354:Topological module
12214:the two functions
12175:
12122:
12090:
12065:such that for all
12055:
12035:
12018:{\displaystyle X,}
12015:
11992:
11968:
11936:
11904:
11875:
11855:
11839:Uniform continuity
11827:{\displaystyle C.}
11824:
11801:
11745:{\displaystyle X.}
11742:
11715:
11691:
11665:
11645:
11633:Neighborhood basis
11613:{\displaystyle X.}
11610:
11587:
11555:
11516:
11492:{\displaystyle X.}
11489:
11466:
11446:
11427:{\displaystyle X.}
11424:
11398:
11375:
11348:
11322:
11284:
11264:
11217:
11197:
11165:
11129:
11109:
11085:
11065:
11033:
11013:
10987:
10946:
10910:
10891:{\displaystyle S.}
10888:
10868:{\displaystyle S.}
10865:
10838:
10818:
10786:
10766:
10749:{\displaystyle S,}
10746:
10723:
10703:
10681:
10659:{\displaystyle S.}
10656:
10633:
10613:
10589:
10565:
10545:
10517:
10499:{\displaystyle S.}
10496:
10473:
10453:
10426:
10406:
10384:{\displaystyle X.}
10381:
10358:
10341:{\displaystyle X.}
10338:
10315:
10295:
10276:{\displaystyle S.}
10273:
10247:
10227:
10180:
10160:
10138:
10096:
10065:
10043:
10019:
9976:
9952:
9916:
9896:
9866:
9837:
9817:
9793:{\displaystyle X.}
9790:
9767:
9755:neighborhood basis
9739:
9704:
9678:
9658:there exists some
9651:{\displaystyle X,}
9648:
9625:
9605:
9581:
9557:
9525:
9495:
9475:
9455:{\displaystyle X,}
9452:
9429:
9409:
9387:
9350:
9305:{\displaystyle X,}
9302:
9279:
9223:
9152:{\displaystyle X,}
9149:
9126:
9079:
9059:
9033:
8997:
8973:
8952:
8935:{\displaystyle 0,}
8932:
8909:
8889:
8869:
8838:
8796:
8750:
8727:there exists some
8720:{\displaystyle X,}
8717:
8694:
8674:
8654:
8555:
8535:
8469:
8355:
8244:
8212:
8182:
8075:
8018:
7950:
7927:{\displaystyle Y.}
7924:
7901:
7840:
7820:
7759:{\displaystyle X,}
7756:
7710:
7690:
7670:
7648:
7574:
7431:
7366:
7330:
7297:
7275:
7246:
7155:
7105:
7081:
7040:
6990:
6953:
6916:
6895:{\displaystyle X,}
6892:
6874:base of entourages
6862:
6802:
6774:{\displaystyle X.}
6771:
6748:
6728:
6708:
6675:
6651:
6614:
6543:
6434:
6411:{\displaystyle 0,}
6408:
6385:
6359:
6296:
6273:{\displaystyle 0.}
6270:
6220:(diffeomorphic to
6172:
6138:
6108:
6038:
6024:graded-commutative
6010:
5897:(which classifies
5843:
5831:, while the group
5814:
5787:
5762:Pontryagin duality
5750:
5708:
5667:
5630:
5608:
5584:
5560:
5479:
5420:PeterβWeyl theorem
5287:
5252:
5217:
5174:
5138:that determines a
5072:
5012:
4972:
4903:
4725:identity component
4703:
4600:
4549:
4494:, the set of left
4422:is also closed in
4372:
4325:
4294:
4267:
4247:
4227:
4207:
4169:(Hausdorff) space.
4155:(Hausdorff) space.
4054:, are pre-compact.
4044:
4018:
3938:
3912:
3876:
3847:
3827:
3775:
3685:
3576:
3488:
3451:
3422:
3304:completely regular
3255:uniform continuity
3175:, where note that
3100:
3029:
2986:
2957:
2937:
2917:
2877:{\displaystyle G.}
2874:
2851:
2792:{\displaystyle G,}
2789:
2766:
2746:
2726:
2709:{\displaystyle G.}
2706:
2683:
2663:
2606:
2577:
2565:neighborhood basis
2553:
2529:
2476:
2426:
2398:
2378:
2349:
2323:
2294:
2190:
2153:
2132:
2090:
2047:
2017:
1978:
1929:
1897:
1850:
1819:
1793:
1748:
1720:
1697:Euclidean geometry
1676:
1637:
1608:
1579:of all invertible
1567:
1496:
1466:
1392:group homomorphism
841:topological spaces
833:topological groups
825:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
18:Topological groups
13350:978-1-4612-7155-0
13202:Montgomery, Deane
13169:Mackey, George W.
13078:978-0-486-68143-6
12941:Bourbaki, Nicolas
12851:978-90-78677-06-2
12807:, pp. 48β51.
12776:, pp. 47β66.
12759:, pp. 12β19.
12627:, section I.11.5.
12520:, pp. 19β45.
12345:Topological field
12058:{\displaystyle Y}
12045:of the origin in
12038:{\displaystyle V}
12002:of the origin in
11995:{\displaystyle U}
11878:{\displaystyle Y}
11858:{\displaystyle X}
11729:of the origin in
11727:neighborhood base
11668:{\displaystyle X}
11648:{\displaystyle C}
11597:to some point of
11558:{\displaystyle X}
11519:{\displaystyle X}
11469:{\displaystyle X}
11449:{\displaystyle 0}
11401:{\displaystyle X}
11378:{\displaystyle X}
11351:{\displaystyle X}
11287:{\displaystyle S}
11220:{\displaystyle S}
11132:{\displaystyle S}
11112:{\displaystyle S}
11088:{\displaystyle S}
11036:{\displaystyle S}
11023:(for example, if
10913:{\displaystyle X}
10841:{\displaystyle S}
10789:{\displaystyle S}
10769:{\displaystyle S}
10726:{\displaystyle S}
10706:{\displaystyle X}
10684:{\displaystyle S}
10636:{\displaystyle S}
10616:{\displaystyle S}
10592:{\displaystyle S}
10568:{\displaystyle X}
10548:{\displaystyle S}
10520:{\displaystyle S}
10476:{\displaystyle S}
10429:{\displaystyle S}
10409:{\displaystyle S}
10361:{\displaystyle X}
10318:{\displaystyle S}
10298:{\displaystyle X}
10250:{\displaystyle S}
10183:{\displaystyle X}
10163:{\displaystyle S}
10068:{\displaystyle S}
9919:{\displaystyle X}
9840:{\displaystyle X}
9770:{\displaystyle 0}
9628:{\displaystyle 0}
9608:{\displaystyle N}
9584:{\displaystyle X}
9498:{\displaystyle N}
9432:{\displaystyle 0}
9412:{\displaystyle N}
9082:{\displaystyle X}
9000:{\displaystyle N}
8976:{\displaystyle N}
8962:is said to be an
8955:{\displaystyle B}
8912:{\displaystyle N}
8892:{\displaystyle X}
8872:{\displaystyle B}
8697:{\displaystyle 0}
8677:{\displaystyle N}
8649:
8558:{\displaystyle X}
8054:
7843:{\displaystyle X}
7735:Net (mathematics)
7693:{\displaystyle X}
7673:{\displaystyle X}
7559:
7530:
7516:
7300:{\displaystyle N}
6852:
6844:
6751:{\displaystyle 0}
6731:{\displaystyle N}
6687:uniform structure
6678:{\displaystyle X}
6528:
6437:{\displaystyle N}
6299:{\displaystyle X}
5944:Whitehead product
5936:fundamental group
5934:For example, the
5911:homotopy category
5890:classifying space
5870:BanachβLie groups
5784:
5747:
5689:, is to find the
5546:Fourier transform
5514:harmonic analysis
4929:
4801:and a closed set
4797:of a compact set
4513:homogeneous space
4509:quotient topology
4490:is a subgroup of
4470:, the closure of
4454:is a subgroup of
4416:subspace topology
4382:is proper. Since
3850:{\displaystyle d}
3605:) if and only if
3579:{\displaystyle G}
3237:in two ways; the
2960:{\displaystyle G}
2940:{\displaystyle G}
2769:{\displaystyle U}
2749:{\displaystyle G}
2736:is any subset of
2729:{\displaystyle S}
2686:{\displaystyle x}
2580:{\displaystyle G}
2429:{\displaystyle G}
2401:{\displaystyle S}
2252:bounded operators
2117:
1778:and then solved.
1437:discrete topology
1420:group isomorphism
1271:Additive notation
908:topological space
900:topological group
894:Formal definition
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
13394:
13387:Fourier analysis
13362:
13328:
13305:(2nd ed.).
13299:Porteous, Ian R.
13294:
13264:
13251:
13224:
13197:
13164:
13136:Ross, Kenneth A.
13127:
13099:Ross, Kenneth A.
13090:
13063:
13028:
12994:
12969:
12936:
12894:
12871:(1st ed.).
12863:
12842:World Scientific
12820:
12814:
12808:
12802:
12789:
12783:
12777:
12771:
12760:
12754:
12748:
12742:
12736:
12730:
12724:
12718:
12712:
12706:
12700:
12694:
12688:
12687:, Theorem 27.40.
12682:
12676:
12670:
12664:
12658:
12652:
12646:
12640:
12634:
12628:
12622:
12616:
12610:
12604:
12603:
12602:
12586:
12580:
12579:
12563:
12557:
12551:
12545:
12539:
12533:
12532:, section III.3.
12527:
12521:
12515:
12482:
12472:
12466:
12460:
12454:
12448:
12432:
12430:
12426:
12419:
12408:
12394:
12373:
12364:
12359:Topological ring
12350:
12341:
12321:
12312:
12280:
12243:
12233:
12223:
12213:
12203:
12184:
12182:
12181:
12176:
12131:
12129:
12128:
12123:
12099:
12097:
12096:
12091:
12064:
12062:
12061:
12056:
12044:
12042:
12041:
12036:
12024:
12022:
12021:
12016:
12001:
11999:
11998:
11993:
11977:
11975:
11974:
11969:
11945:
11943:
11942:
11937:
11913:
11911:
11910:
11905:
11884:
11882:
11881:
11876:
11864:
11862:
11861:
11856:
11841:
11840:
11833:
11831:
11830:
11825:
11810:
11808:
11807:
11802:
11800:
11796:
11795:
11794:
11773:
11772:
11751:
11749:
11748:
11743:
11724:
11722:
11721:
11716:
11714:
11713:
11700:
11698:
11697:
11692:
11674:
11672:
11671:
11666:
11654:
11652:
11651:
11646:
11619:
11617:
11616:
11611:
11596:
11594:
11593:
11588:
11564:
11562:
11561:
11556:
11525:
11523:
11522:
11517:
11498:
11496:
11495:
11490:
11475:
11473:
11472:
11467:
11455:
11453:
11452:
11447:
11433:
11431:
11430:
11425:
11407:
11405:
11404:
11399:
11384:
11382:
11381:
11376:
11357:
11355:
11354:
11349:
11331:
11329:
11328:
11323:
11293:
11291:
11290:
11285:
11273:
11271:
11270:
11265:
11226:
11224:
11223:
11218:
11207:that are not in
11206:
11204:
11203:
11198:
11174:
11172:
11171:
11166:
11139:), converges to
11138:
11136:
11135:
11130:
11118:
11116:
11115:
11110:
11094:
11092:
11091:
11086:
11074:
11072:
11071:
11066:
11042:
11040:
11039:
11034:
11022:
11020:
11019:
11014:
10996:
10994:
10993:
10988:
10955:
10953:
10952:
10947:
10919:
10917:
10916:
10911:
10897:
10895:
10894:
10889:
10874:
10872:
10871:
10866:
10847:
10845:
10844:
10839:
10827:
10825:
10824:
10819:
10795:
10793:
10792:
10787:
10775:
10773:
10772:
10767:
10755:
10753:
10752:
10747:
10732:
10730:
10729:
10724:
10712:
10710:
10709:
10704:
10690:
10688:
10687:
10682:
10665:
10663:
10662:
10657:
10642:
10640:
10639:
10634:
10622:
10620:
10619:
10614:
10598:
10596:
10595:
10590:
10574:
10572:
10571:
10566:
10554:
10552:
10551:
10546:
10526:
10524:
10523:
10518:
10505:
10503:
10502:
10497:
10482:
10480:
10479:
10474:
10462:
10460:
10459:
10454:
10452:
10451:
10435:
10433:
10432:
10427:
10415:
10413:
10412:
10407:
10390:
10388:
10387:
10382:
10367:
10365:
10364:
10359:
10347:
10345:
10344:
10339:
10324:
10322:
10321:
10316:
10304:
10302:
10301:
10296:
10282:
10280:
10279:
10274:
10256:
10254:
10253:
10248:
10236:
10234:
10233:
10228:
10211:
10210:
10189:
10187:
10186:
10181:
10169:
10167:
10166:
10161:
10147:
10145:
10144:
10139:
10119:
10118:
10105:
10103:
10102:
10097:
10074:
10072:
10071:
10066:
10052:
10050:
10049:
10044:
10042:
10041:
10028:
10026:
10025:
10020:
9985:
9983:
9982:
9977:
9975:
9974:
9961:
9959:
9958:
9953:
9951:
9950:
9925:
9923:
9922:
9917:
9905:
9903:
9902:
9897:
9889:
9888:
9875:
9873:
9872:
9867:
9846:
9844:
9843:
9838:
9826:
9824:
9823:
9818:
9816:
9815:
9799:
9797:
9796:
9791:
9776:
9774:
9773:
9768:
9748:
9746:
9745:
9740:
9713:
9711:
9710:
9705:
9687:
9685:
9684:
9679:
9677:
9676:
9657:
9655:
9654:
9649:
9634:
9632:
9631:
9626:
9614:
9612:
9611:
9606:
9590:
9588:
9587:
9582:
9566:
9564:
9563:
9558:
9534:
9532:
9531:
9526:
9524:
9523:
9504:
9502:
9501:
9496:
9484:
9482:
9481:
9476:
9474:
9473:
9461:
9459:
9458:
9453:
9438:
9436:
9435:
9430:
9418:
9416:
9415:
9410:
9396:
9394:
9393:
9388:
9383:
9382:
9373:
9372:
9359:
9357:
9356:
9351:
9346:
9345:
9311:
9309:
9308:
9303:
9288:
9286:
9285:
9280:
9269:
9268:
9232:
9230:
9229:
9224:
9219:
9218:
9182:
9181:
9172:
9171:
9158:
9156:
9155:
9150:
9135:
9133:
9132:
9127:
9119:
9118:
9109:
9108:
9091:Cauchy prefilter
9088:
9086:
9085:
9080:
9068:
9066:
9065:
9060:
9058:
9057:
9042:
9040:
9039:
9034:
9006:
9004:
9003:
8998:
8982:
8980:
8979:
8974:
8961:
8959:
8958:
8953:
8941:
8939:
8938:
8933:
8918:
8916:
8915:
8910:
8898:
8896:
8895:
8890:
8878:
8876:
8875:
8870:
8847:
8845:
8844:
8839:
8834:
8833:
8806:for all indices
8805:
8803:
8802:
8797:
8789:
8788:
8776:
8775:
8759:
8757:
8756:
8751:
8743:
8742:
8726:
8724:
8723:
8718:
8703:
8701:
8700:
8695:
8683:
8681:
8680:
8675:
8663:
8661:
8660:
8655:
8650:
8647:
8639:
8638:
8609:
8605:
8604:
8603:
8591:
8590:
8564:
8562:
8561:
8556:
8544:
8542:
8541:
8536:
8534:
8533:
8522:
8518:
8517:
8500:
8499:
8478:
8476:
8475:
8470:
8465:
8464:
8435:
8431:
8430:
8429:
8417:
8416:
8398:
8397:
8385:
8384:
8364:
8362:
8361:
8356:
8354:
8353:
8324:
8320:
8319:
8318:
8306:
8305:
8287:
8286:
8274:
8273:
8253:
8251:
8250:
8245:
8221:
8219:
8218:
8213:
8191:
8189:
8188:
8183:
8181:
8180:
8151:
8147:
8146:
8145:
8133:
8132:
8114:
8113:
8101:
8100:
8084:
8082:
8081:
8076:
8071:
8070:
8055:
8052:
8050:
8049:
8027:
8025:
8024:
8019:
8017:
8013:
8012:
8011:
7999:
7998:
7959:
7957:
7956:
7951:
7933:
7931:
7930:
7925:
7910:
7908:
7907:
7902:
7900:
7899:
7888:
7884:
7883:
7866:
7865:
7849:
7847:
7846:
7841:
7829:
7827:
7826:
7821:
7819:
7818:
7807:
7803:
7802:
7785:
7784:
7765:
7763:
7762:
7757:
7719:
7717:
7716:
7711:
7699:
7697:
7696:
7691:
7679:
7677:
7676:
7671:
7657:
7655:
7654:
7649:
7626:
7625:
7573:
7531:
7528:
7517:
7514:
7479:
7478:
7457:
7456:
7440:
7438:
7437:
7432:
7421:
7420:
7408:
7407:
7392:
7391:
7375:
7373:
7372:
7367:
7356:
7355:
7339:
7337:
7336:
7331:
7306:
7304:
7303:
7298:
7284:
7282:
7281:
7276:
7255:
7253:
7252:
7247:
7194:
7193:
7181:
7180:
7164:
7162:
7161:
7156:
7145:
7144:
7129:Every entourage
7114:
7112:
7111:
7106:
7104:
7103:
7090:
7088:
7087:
7082:
7049:
7047:
7046:
7041:
6999:
6997:
6996:
6991:
6962:
6960:
6959:
6954:
6949:
6948:
6925:
6923:
6922:
6917:
6915:
6914:
6901:
6899:
6898:
6893:
6871:
6869:
6868:
6863:
6861:
6857:
6853:
6850:
6845:
6842:
6811:
6809:
6808:
6803:
6780:
6778:
6777:
6772:
6757:
6755:
6754:
6749:
6737:
6735:
6734:
6729:
6717:
6715:
6714:
6709:
6684:
6682:
6681:
6676:
6660:
6658:
6657:
6652:
6623:
6621:
6620:
6615:
6589:
6588:
6542:
6461:
6460:
6443:
6441:
6440:
6435:
6417:
6415:
6414:
6409:
6394:
6392:
6391:
6386:
6368:
6366:
6365:
6360:
6322:
6321:
6305:
6303:
6302:
6297:
6279:
6277:
6276:
6271:
6227:
6219:
6212:
6207:to the 3-sphere
6183:
6181:
6179:
6178:
6173:
6171:
6152:hyperbolic plane
6149:
6147:
6145:
6144:
6139:
6137:
6123:
6119:
6117:
6115:
6114:
6109:
6107:
6089:
6074:
6070:
6066:
6058:exterior algebra
6047:
6045:
6044:
6039:
6037:
6021:
6019:
6017:
6016:
6011:
6009:
5988:
5972:
5949:
5941:
5926:
5922:
5908:
5902:
5896:
5887:
5863:
5859:
5852:
5850:
5849:
5844:
5842:
5830:
5824:
5823:
5821:
5820:
5815:
5813:
5800:
5796:
5794:
5793:
5788:
5786:
5785:
5777:
5767:
5759:
5757:
5756:
5751:
5749:
5748:
5740:
5730:
5720:
5717:
5715:
5714:
5709:
5707:
5677:
5676:
5674:
5673:
5668:
5666:
5649:
5639:
5637:
5636:
5631:
5629:
5617:
5615:
5614:
5609:
5607:
5595:
5593:
5591:
5590:
5585:
5583:
5569:
5567:
5566:
5561:
5559:
5535:
5497:
5489:
5488:
5486:
5485:
5480:
5478:
5465:
5459:
5455:
5445:
5439:
5431:
5394:
5384:
5370:
5362:
5352:
5334:
5322:
5306:
5302:
5296:
5294:
5293:
5288:
5286:
5268:
5261:
5259:
5258:
5253:
5251:
5233:
5226:
5224:
5223:
5218:
5216:
5189:
5183:
5181:
5180:
5175:
5173:
5145:
5133:
5125:
5116:Deane Montgomery
5109:
5101:
5084:Cartan's theorem
5081:
5079:
5078:
5073:
5044:
5040:
5032:
5021:
5019:
5018:
5013:
4981:
4979:
4978:
4973:
4962:
4942:
4931:
4930:
4922:
4912:
4910:
4909:
4904:
4865:
4862:is closed, then
4861:
4851:
4847:
4843:
4839:
4832:
4816:
4812:
4808:
4804:
4800:
4796:
4778:
4768:
4764:
4760:
4752:
4748:
4742:
4739:is any point of
4734:
4719:
4713:
4712:
4710:
4709:
4704:
4702:
4697:
4692:
4679:
4675:
4671:
4658:
4650:
4643:
4639:
4635:
4625:
4610:
4609:
4607:
4606:
4601:
4599:
4586:
4575:
4566:
4558:
4556:
4555:
4550:
4545:
4518:
4506:
4493:
4489:
4477:
4473:
4469:
4465:
4461:
4457:
4453:
4449:
4435:
4429:
4425:
4421:
4401:
4397:
4392:second countable
4389:
4385:
4381:
4379:
4378:
4373:
4371:
4370:
4354:
4350:
4342:
4334:
4332:
4331:
4326:
4324:
4323:
4303:
4301:
4300:
4295:
4276:
4274:
4273:
4268:
4256:
4254:
4253:
4248:
4236:
4234:
4233:
4228:
4216:
4214:
4213:
4208:
4189:
4185:
4174:
4160:
4150:second countable
4147:
4140:
4132:
4128:
4121:
4117:
4106:
4100:
4092:
4080:
4074:are equivalent:
4073:
4064:Garrett Birkhoff
4053:
4051:
4050:
4045:
4027:
4025:
4024:
4019:
3947:
3945:
3944:
3939:
3921:
3919:
3918:
3913:
3885:
3883:
3882:
3877:
3856:
3854:
3853:
3848:
3836:
3834:
3833:
3828:
3820:
3819:
3807:
3806:
3784:
3782:
3781:
3776:
3771:
3770:
3758:
3757:
3733:
3732:
3717:
3716:
3694:
3692:
3691:
3686:
3681:
3680:
3668:
3667:
3646:
3645:
3630:
3629:
3593:
3589:
3585:
3583:
3582:
3577:
3565:
3561:
3553:
3549:
3537:
3525:
3521:
3511:
3497:
3495:
3494:
3489:
3468:
3464:
3460:
3458:
3457:
3452:
3431:
3429:
3428:
3423:
3402:
3398:
3394:
3390:
3389:
3388:
3387:
3383:
3382:
3371:
3367:
3356:
3355:
3351:
3344:
3330:
3316:
3309:
3298:
3288:
3284:
3280:
3276:
3272:
3248:
3243:right uniformity
3224:
3217:
3206:
3202:
3198:
3190:
3183:
3174:
3161:
3157:
3153:
3146:
3137:
3127:
3117:
3113:
3109:
3107:
3106:
3101:
3096:
3092:
3079:
3078:
3058:
3057:
3038:
3036:
3035:
3030:
3019:
3018:
2995:
2993:
2992:
2987:
2966:
2964:
2963:
2958:
2946:
2944:
2943:
2938:
2926:
2924:
2923:
2918:
2916:
2915:
2883:
2881:
2880:
2875:
2860:
2858:
2857:
2852:
2798:
2796:
2795:
2790:
2775:
2773:
2772:
2767:
2755:
2753:
2752:
2747:
2735:
2733:
2732:
2727:
2715:
2713:
2712:
2707:
2692:
2690:
2689:
2684:
2672:
2670:
2669:
2664:
2659:
2658:
2631:
2630:
2615:
2613:
2612:
2607:
2586:
2584:
2583:
2578:
2562:
2560:
2559:
2554:
2552:
2551:
2538:
2536:
2535:
2530:
2485:
2483:
2482:
2477:
2435:
2433:
2432:
2427:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2358:
2356:
2355:
2350:
2332:
2330:
2329:
2324:
2303:
2301:
2300:
2295:
2273:
2272:
2229:KacβMoody groups
2199:
2197:
2196:
2191:
2189:
2188:
2179:
2174:
2162:
2160:
2159:
2154:
2152:
2151:
2142:
2137:
2131:
2108:pro-finite group
2105:
2099:
2097:
2096:
2091:
2089:
2062:
2056:
2054:
2053:
2048:
2046:
2033:
2026:
2024:
2023:
2018:
2016:
1994:
1987:
1985:
1984:
1979:
1977:
1958:-adic Lie groups
1944:
1938:
1936:
1935:
1930:
1928:
1911:
1906:
1904:
1903:
1898:
1896:
1879:
1865:
1859:
1857:
1856:
1851:
1849:
1828:
1826:
1825:
1820:
1818:
1805:rational numbers
1802:
1800:
1799:
1794:
1792:
1768:smooth manifolds
1758:
1757:
1755:
1754:
1749:
1747:
1730:
1729:
1727:
1726:
1721:
1719:
1686:
1685:
1683:
1682:
1677:
1675:
1658:
1650:orthogonal group
1647:
1646:
1644:
1643:
1638:
1636:
1619:
1617:
1615:
1614:
1609:
1607:
1586:
1582:
1578:
1576:
1574:
1573:
1568:
1566:
1544:classical groups
1538:
1534:
1523:
1506:
1505:
1503:
1502:
1497:
1495:
1481:
1475:
1473:
1472:
1467:
1465:
1402:
1376:category of sets
1346:
1333:
1323:
1311:
1292:
1266:
1252:
1248:
1244:
1240:
1236:
1232:
1226:
1222:
1218:
1201:
1186:
1166:
1148:
1138:
1134:
1130:
1126:
1122:
1116:
1112:
1102:
1068:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1024:
1020:
995:and is called a
989:product topology
986:
969:
959:
942:
927:
905:
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
21:
13402:
13401:
13397:
13396:
13395:
13393:
13392:
13391:
13367:
13366:
13365:
13351:
13317:
13283:
13240:
13187:
13154:
13144:Springer-Verlag
13142:, vol. 2,
13117:
13107:Springer-Verlag
13079:
13053:
13017:
13007:Springer-Verlag
12999:Bredon, Glen E.
12992:
12959:
12949:Springer-Verlag
12883:
12873:Springer-Verlag
12852:
12828:
12823:
12815:
12811:
12803:
12792:
12784:
12780:
12772:
12763:
12755:
12751:
12743:
12739:
12735:, Theorem 3C.4.
12731:
12727:
12723:, Theorem 4.66.
12719:
12715:
12709:Banaszczyk 1983
12707:
12703:
12695:
12691:
12683:
12679:
12671:
12667:
12659:
12655:
12651:, section 4.10.
12647:
12643:
12635:
12631:
12623:
12619:
12611:
12607:
12587:
12583:
12564:
12560:
12556:, section 1.22.
12552:
12548:
12540:
12536:
12528:
12524:
12516:
12485:
12473:
12469:
12461:
12457:
12451:Pontrjagin 1946
12449:
12445:
12441:
12436:
12435:
12428:
12421:
12410:
12400:
12395:
12391:
12386:
12371:
12362:
12348:
12339:
12325:Profinite group
12319:
12310:
12278:
12269:Algebraic group
12265:
12244:are continuous.
12235:
12225:
12215:
12205:
12201:
12190:
12188:Generalizations
12137:
12134:
12133:
12105:
12102:
12101:
12070:
12067:
12066:
12050:
12047:
12046:
12030:
12027:
12026:
12007:
12004:
12003:
11987:
11984:
11983:
11951:
11948:
11947:
11946:be a map. Then
11919:
11916:
11915:
11890:
11887:
11886:
11870:
11867:
11866:
11850:
11847:
11846:
11838:
11837:
11816:
11813:
11812:
11790:
11789:
11768:
11764:
11763:
11759:
11757:
11754:
11753:
11734:
11731:
11730:
11709:
11708:
11706:
11703:
11702:
11680:
11677:
11676:
11660:
11657:
11656:
11640:
11637:
11636:
11602:
11599:
11598:
11570:
11567:
11566:
11550:
11547:
11546:
11541:if each Cauchy
11511:
11508:
11507:
11481:
11478:
11477:
11461:
11458:
11457:
11441:
11438:
11437:
11416:
11413:
11412:
11393:
11390:
11389:
11370:
11367:
11366:
11343:
11340:
11339:
11299:
11296:
11295:
11279:
11276:
11275:
11235:
11232:
11231:
11212:
11209:
11208:
11180:
11177:
11176:
11148:
11145:
11144:
11124:
11121:
11120:
11104:
11101:
11100:
11080:
11077:
11076:
11048:
11045:
11044:
11028:
11025:
11024:
11002:
10999:
10998:
10961:
10958:
10957:
10932:
10929:
10928:
10905:
10902:
10901:
10880:
10877:
10876:
10857:
10854:
10853:
10833:
10830:
10829:
10804:
10801:
10800:
10781:
10778:
10777:
10761:
10758:
10757:
10738:
10735:
10734:
10718:
10715:
10714:
10698:
10695:
10694:
10676:
10673:
10672:
10648:
10645:
10644:
10628:
10625:
10624:
10608:
10605:
10604:
10584:
10581:
10580:
10560:
10557:
10556:
10540:
10537:
10536:
10512:
10509:
10508:
10488:
10485:
10484:
10468:
10465:
10464:
10447:
10446:
10444:
10441:
10440:
10421:
10418:
10417:
10401:
10398:
10397:
10373:
10370:
10369:
10353:
10350:
10349:
10330:
10327:
10326:
10310:
10307:
10306:
10290:
10287:
10286:
10265:
10262:
10261:
10242:
10239:
10238:
10206:
10205:
10203:
10200:
10199:
10192:complete subset
10175:
10172:
10171:
10155:
10152:
10151:
10114:
10113:
10111:
10108:
10107:
10082:
10079:
10078:
10060:
10057:
10056:
10037:
10036:
10034:
10031:
10030:
10005:
10002:
10001:
9998:
9992:
9970:
9969:
9967:
9964:
9963:
9946:
9945:
9931:
9928:
9927:
9926:if and only if
9911:
9908:
9907:
9884:
9883:
9881:
9878:
9877:
9852:
9849:
9848:
9832:
9829:
9828:
9811:
9810:
9808:
9805:
9804:
9782:
9779:
9778:
9762:
9759:
9758:
9719:
9716:
9715:
9693:
9690:
9689:
9672:
9671:
9663:
9660:
9659:
9640:
9637:
9636:
9620:
9617:
9616:
9600:
9597:
9596:
9576:
9573:
9572:
9540:
9537:
9536:
9519:
9518:
9510:
9507:
9506:
9490:
9487:
9486:
9469:
9468:
9466:
9463:
9462:
9444:
9441:
9440:
9424:
9421:
9420:
9404:
9401:
9400:
9378:
9377:
9368:
9367:
9365:
9362:
9361:
9341:
9340:
9317:
9314:
9313:
9294:
9291:
9290:
9264:
9263:
9240:
9237:
9236:
9233:is a prefilter.
9214:
9213:
9177:
9176:
9167:
9166:
9164:
9161:
9160:
9141:
9138:
9137:
9114:
9113:
9104:
9103:
9101:
9098:
9097:
9074:
9071:
9070:
9053:
9052:
9050:
9047:
9046:
9013:
9010:
9009:
8992:
8989:
8988:
8987:small of order
8968:
8965:
8964:
8947:
8944:
8943:
8924:
8921:
8920:
8904:
8901:
8900:
8884:
8881:
8880:
8864:
8861:
8860:
8853:Cauchy sequence
8829:
8825:
8811:
8808:
8807:
8784:
8780:
8771:
8767:
8765:
8762:
8761:
8738:
8734:
8732:
8729:
8728:
8709:
8706:
8705:
8689:
8686:
8685:
8669:
8666:
8665:
8646:
8610:
8599:
8595:
8586:
8582:
8581:
8577:
8576:
8574:
8571:
8570:
8550:
8547:
8546:
8523:
8513:
8509:
8505:
8504:
8495:
8491:
8489:
8486:
8485:
8436:
8425:
8421:
8412:
8408:
8407:
8403:
8402:
8393:
8389:
8380:
8376:
8374:
8371:
8370:
8325:
8314:
8310:
8301:
8297:
8296:
8292:
8291:
8282:
8278:
8269:
8265:
8263:
8260:
8259:
8227:
8224:
8223:
8201:
8198:
8197:
8152:
8141:
8137:
8128:
8124:
8123:
8119:
8118:
8109:
8105:
8096:
8092:
8090:
8087:
8086:
8066:
8062:
8053: and
8051:
8045:
8041:
8033:
8030:
8029:
8028:if and only if
8007:
8003:
7994:
7990:
7989:
7985:
7965:
7962:
7961:
7939:
7936:
7935:
7916:
7913:
7912:
7889:
7879:
7875:
7871:
7870:
7861:
7857:
7855:
7852:
7851:
7835:
7832:
7831:
7808:
7798:
7794:
7790:
7789:
7780:
7776:
7774:
7771:
7770:
7748:
7745:
7744:
7737:
7729:Main articles:
7727:
7705:
7702:
7701:
7685:
7682:
7681:
7665:
7662:
7661:
7621:
7617:
7563:
7527:
7513:
7474:
7470:
7452:
7448:
7446:
7443:
7442:
7416:
7412:
7403:
7399:
7387:
7383:
7381:
7378:
7377:
7351:
7347:
7345:
7342:
7341:
7316:
7313:
7312:
7292:
7289:
7288:
7261:
7258:
7257:
7189:
7185:
7176:
7172:
7170:
7167:
7166:
7140:
7136:
7134:
7131:
7130:
7099:
7098:
7096:
7093:
7092:
7055:
7052:
7051:
7005:
7002:
7001:
7000:if and only if
6967:
6964:
6963:
6944:
6943:
6935:
6932:
6931:
6910:
6909:
6907:
6904:
6903:
6884:
6881:
6880:
6849:
6841:
6822:
6818:
6816:
6813:
6812:
6788:
6785:
6784:
6763:
6760:
6759:
6743:
6740:
6739:
6723:
6720:
6719:
6694:
6691:
6690:
6670:
6667:
6666:
6631:
6628:
6627:
6584:
6580:
6532:
6456:
6452:
6450:
6447:
6446:
6429:
6426:
6425:
6400:
6397:
6396:
6374:
6371:
6370:
6317:
6313:
6311:
6308:
6307:
6291:
6288:
6287:
6265:
6262:
6261:
6258:
6252:
6240:
6234:
6221:
6214:
6208:
6189:Wilhelm Killing
6167:
6165:
6162:
6161:
6159:
6133:
6131:
6128:
6127:
6125:
6121:
6103:
6101:
6098:
6097:
6095:
6087:
6072:
6068:
6064:
6054:polynomial ring
6048:, that is, the
6033:
6031:
6028:
6027:
6005:
6003:
6000:
5999:
5990:
5986:
5959:
5947:
5939:
5924:
5918:
5906:
5900:
5892:
5885:
5878:
5861:
5857:
5838:
5836:
5833:
5832:
5826:
5809:
5807:
5804:
5803:
5802:
5798:
5776:
5775:
5773:
5770:
5769:
5765:
5739:
5738:
5736:
5733:
5732:
5728:
5721:, but not all.
5703:
5701:
5698:
5697:
5694:
5662:
5660:
5657:
5656:
5651:
5641:
5625:
5623:
5620:
5619:
5603:
5601:
5598:
5597:
5579:
5577:
5574:
5573:
5571:
5555:
5553:
5550:
5549:
5533:
5530:direct integral
5524:, given by the
5491:
5474:
5472:
5469:
5468:
5467:
5461:
5457:
5447:
5441:
5433:
5427:
5386:
5376:
5368:
5360:
5340:
5332:
5320:
5313:
5304:
5282:
5280:
5277:
5276:
5270:
5266:
5247:
5245:
5242:
5241:
5235:
5231:
5212:
5210:
5207:
5206:
5200:
5188:
5169:
5167:
5164:
5163:
5162:
5159:profinite group
5148:covering spaces
5143:
5134:, an object of
5131:
5123:
5107:
5099:
5061:
5058:
5057:
5054:
5042:
5034:
5023:
4995:
4992:
4991:
4955:
4938:
4921:
4920:
4918:
4915:
4914:
4886:
4883:
4882:
4872:
4863:
4853:
4849:
4845:
4841:
4837:
4818:
4814:
4810:
4806:
4802:
4798:
4792:
4789:
4770:
4766:
4762:
4758:
4750:
4744:
4740:
4732:
4715:
4698:
4693:
4688:
4686:
4683:
4682:
4681:
4677:
4673:
4663:
4656:
4653:normal subgroup
4648:
4641:
4637:
4627:
4612:
4595:
4593:
4590:
4589:
4588:
4580:
4571:
4564:
4541:
4524:
4521:
4520:
4516:
4498:
4491:
4487:
4484:
4475:
4471:
4467:
4463:
4459:
4455:
4451:
4437:
4431:
4427:
4423:
4419:
4408:
4399:
4395:
4387:
4383:
4366:
4362:
4360:
4357:
4356:
4352:
4348:
4340:
4319:
4315:
4313:
4310:
4309:
4289:
4286:
4285:
4262:
4259:
4258:
4242:
4239:
4238:
4222:
4219:
4218:
4202:
4199:
4198:
4187:
4183:
4172:
4167:locally compact
4158:
4153:locally compact
4145:
4138:
4130:
4126:
4119:
4115:
4104:
4098:
4090:
4087:first countable
4078:
4071:
4068:Shizuo Kakutani
4033:
4030:
4029:
3959:
3956:
3955:
3927:
3924:
3923:
3895:
3892:
3891:
3862:
3859:
3858:
3842:
3839:
3838:
3837:(equivalently,
3815:
3811:
3802:
3798:
3790:
3787:
3786:
3766:
3762:
3753:
3749:
3728:
3724:
3712:
3708:
3700:
3697:
3696:
3676:
3672:
3663:
3659:
3641:
3637:
3625:
3621:
3610:
3607:
3606:
3603:right-invariant
3591:
3587:
3571:
3568:
3567:
3563:
3559:
3551:
3547:
3544:
3535:
3523:
3513:
3509:
3474:
3471:
3470:
3466:
3462:
3437:
3434:
3433:
3411:
3408:
3407:
3400:
3396:
3385:
3384:
3380:
3379:
3378:
3377:
3375:
3369:
3365:
3357:
3353:
3349:
3348:
3342:
3334:
3328:
3320:
3314:
3307:
3290:
3286:
3282:
3278:
3274:
3270:
3267:
3246:
3239:left uniformity
3231:
3219:
3208:
3204:
3200:
3196:
3193:locally compact
3188:
3176:
3163:
3159:
3155:
3151:
3139:
3129:
3119:
3115:
3111:
3071:
3067:
3066:
3062:
3050:
3046:
3044:
3041:
3040:
3011:
3007:
3005:
3002:
3001:
2975:
2972:
2971:
2952:
2949:
2948:
2932:
2929:
2928:
2908:
2904:
2896:
2893:
2892:
2889:
2866:
2863:
2862:
2804:
2801:
2800:
2781:
2778:
2777:
2761:
2758:
2757:
2741:
2738:
2737:
2721:
2718:
2717:
2698:
2695:
2694:
2678:
2675:
2674:
2654:
2653:
2626:
2625:
2620:
2617:
2616:
2592:
2589:
2588:
2572:
2569:
2568:
2547:
2546:
2544:
2541:
2540:
2491:
2488:
2487:
2441:
2438:
2437:
2421:
2418:
2417:
2393:
2390:
2389:
2364:
2361:
2360:
2338:
2335:
2334:
2309:
2306:
2305:
2280:
2277:
2276:
2270:
2269:
2265:
2260:
2184:
2180:
2175:
2170:
2168:
2165:
2164:
2147:
2143:
2138:
2133:
2121:
2115:
2112:
2111:
2104:
2085:
2083:
2080:
2079:
2078:
2061:
2042:
2040:
2037:
2036:
2035:
2031:
2012:
2010:
2007:
2006:
2000:
1992:
1973:
1971:
1968:
1967:
1961:
1943:
1924:
1922:
1919:
1918:
1917:
1892:
1890:
1887:
1886:
1885:
1877:
1864:
1845:
1843:
1840:
1839:
1838:
1814:
1812:
1809:
1808:
1788:
1786:
1783:
1782:
1743:
1741:
1738:
1737:
1736:
1715:
1713:
1710:
1709:
1700:
1671:
1669:
1666:
1665:
1664:
1652:
1632:
1630:
1627:
1626:
1625:
1603:
1601:
1598:
1597:
1591:
1584:
1580:
1562:
1560:
1557:
1556:
1550:
1536:
1528:
1519:
1491:
1489:
1486:
1485:
1484:
1479:
1461:
1459:
1456:
1455:
1441:discrete groups
1433:
1422:that is also a
1394:
1384:
1364:category theory
1344:
1325:
1314:
1294:
1279:
1254:
1250:
1246:
1242:
1238:
1234:
1228:
1224:
1220:
1206:
1188:
1174:
1150:
1140:
1136:
1132:
1128:
1124:
1118:
1114:
1104:
1070:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1026:
1022:
1008:
978:
961:
950:
929:
914:
903:
896:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
13400:
13390:
13389:
13384:
13379:
13364:
13363:
13349:
13329:
13315:
13295:
13281:
13265:
13252:
13239:978-1584888666
13238:
13225:
13198:
13185:
13165:
13153:978-0387048321
13152:
13128:
13116:978-0387941905
13115:
13091:
13077:
13064:
13051:
13033:Hatcher, Allen
13029:
13015:
12995:
12990:
12970:
12957:
12937:
12912:(4): 485β493,
12895:
12881:
12869:Basic Topology
12864:
12850:
12829:
12827:
12824:
12822:
12821:
12809:
12790:
12778:
12761:
12749:
12737:
12725:
12713:
12701:
12699:, section 2.4.
12689:
12677:
12665:
12663:, section 4.6.
12653:
12641:
12629:
12617:
12605:
12600:10.1.1.236.827
12581:
12558:
12546:
12534:
12522:
12483:
12477:, p. 73;
12475:Armstrong 1997
12467:
12455:
12442:
12440:
12437:
12434:
12433:
12388:
12387:
12385:
12382:
12381:
12380:
12374:
12365:
12356:
12351:
12342:
12333:
12328:
12322:
12313:
12304:
12299:
12293:
12287:
12281:
12275:Complete field
12272:
12264:
12261:
12260:
12259:
12252:
12245:
12189:
12186:
12174:
12171:
12168:
12165:
12162:
12159:
12156:
12153:
12150:
12147:
12144:
12141:
12121:
12118:
12115:
12112:
12109:
12089:
12086:
12083:
12080:
12077:
12074:
12054:
12034:
12014:
12011:
11991:
11967:
11964:
11961:
11958:
11955:
11935:
11932:
11929:
11926:
11923:
11903:
11900:
11897:
11894:
11874:
11854:
11823:
11820:
11799:
11793:
11788:
11785:
11782:
11779:
11776:
11771:
11767:
11762:
11741:
11738:
11712:
11690:
11687:
11684:
11664:
11644:
11623:
11622:
11621:
11620:
11609:
11606:
11586:
11583:
11580:
11577:
11574:
11554:
11535:uniform spaces
11515:
11504:
11503:
11502:
11488:
11485:
11465:
11445:
11434:
11423:
11420:
11397:
11386:
11374:
11360:complete group
11347:
11336:
11335:
11334:
11333:
11321:
11318:
11315:
11312:
11309:
11306:
11303:
11283:
11263:
11260:
11257:
11254:
11251:
11248:
11245:
11242:
11239:
11228:
11216:
11196:
11193:
11190:
11187:
11184:
11164:
11161:
11158:
11155:
11152:
11128:
11108:
11099:Cauchy net in
11084:
11064:
11061:
11058:
11055:
11052:
11032:
11012:
11009:
11006:
10986:
10983:
10980:
10977:
10974:
10971:
10968:
10965:
10945:
10942:
10939:
10936:
10909:
10887:
10884:
10864:
10861:
10837:
10817:
10814:
10811:
10808:
10785:
10765:
10745:
10742:
10722:
10702:
10680:
10655:
10652:
10632:
10612:
10588:
10577:
10576:
10564:
10544:
10516:
10506:
10495:
10492:
10472:
10450:
10437:
10425:
10405:
10394:
10393:
10392:
10380:
10377:
10357:
10337:
10334:
10314:
10294:
10272:
10269:
10246:
10226:
10223:
10220:
10217:
10214:
10209:
10179:
10159:
10137:
10134:
10131:
10128:
10125:
10122:
10117:
10095:
10092:
10089:
10086:
10064:
10040:
10018:
10015:
10012:
10009:
9994:Main article:
9991:
9988:
9973:
9949:
9944:
9941:
9938:
9935:
9915:
9895:
9892:
9887:
9865:
9862:
9859:
9856:
9836:
9814:
9801:
9800:
9789:
9786:
9766:
9750:
9749:
9738:
9735:
9732:
9729:
9726:
9723:
9703:
9700:
9697:
9675:
9670:
9667:
9647:
9644:
9624:
9604:
9580:
9569:
9568:
9556:
9553:
9550:
9547:
9544:
9522:
9517:
9514:
9494:
9485:contains some
9472:
9451:
9448:
9428:
9408:
9397:
9386:
9381:
9376:
9371:
9349:
9344:
9339:
9336:
9333:
9330:
9327:
9324:
9321:
9301:
9298:
9278:
9275:
9272:
9267:
9262:
9259:
9256:
9253:
9250:
9247:
9244:
9234:
9222:
9217:
9212:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9188:
9185:
9180:
9175:
9170:
9148:
9145:
9125:
9122:
9117:
9112:
9107:
9078:
9056:
9032:
9029:
9026:
9023:
9020:
9017:
8996:
8972:
8951:
8931:
8928:
8908:
8888:
8868:
8837:
8832:
8828:
8824:
8821:
8818:
8815:
8795:
8792:
8787:
8783:
8779:
8774:
8770:
8749:
8746:
8741:
8737:
8716:
8713:
8693:
8673:
8653:
8648: in
8645:
8642:
8637:
8634:
8631:
8628:
8625:
8622:
8619:
8616:
8613:
8608:
8602:
8598:
8594:
8589:
8585:
8580:
8554:
8532:
8529:
8526:
8521:
8516:
8512:
8508:
8503:
8498:
8494:
8468:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8442:
8439:
8434:
8428:
8424:
8420:
8415:
8411:
8406:
8401:
8396:
8392:
8388:
8383:
8379:
8352:
8349:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8323:
8317:
8313:
8309:
8304:
8300:
8295:
8290:
8285:
8281:
8277:
8272:
8268:
8243:
8240:
8237:
8234:
8231:
8211:
8208:
8205:
8179:
8176:
8173:
8170:
8167:
8164:
8161:
8158:
8155:
8150:
8144:
8140:
8136:
8131:
8127:
8122:
8117:
8112:
8108:
8104:
8099:
8095:
8074:
8069:
8065:
8061:
8058:
8048:
8044:
8040:
8037:
8016:
8010:
8006:
8002:
7997:
7993:
7988:
7984:
7981:
7978:
7975:
7972:
7969:
7949:
7946:
7943:
7923:
7920:
7898:
7895:
7892:
7887:
7882:
7878:
7874:
7869:
7864:
7860:
7839:
7817:
7814:
7811:
7806:
7801:
7797:
7793:
7788:
7783:
7779:
7755:
7752:
7741:uniform spaces
7726:
7723:
7722:
7721:
7709:
7689:
7669:
7658:
7647:
7644:
7641:
7638:
7635:
7632:
7629:
7624:
7620:
7616:
7613:
7610:
7607:
7604:
7601:
7598:
7595:
7592:
7589:
7586:
7583:
7580:
7577:
7572:
7569:
7566:
7562:
7558:
7555:
7552:
7549:
7546:
7543:
7540:
7537:
7534:
7526:
7523:
7520:
7512:
7509:
7506:
7503:
7500:
7497:
7494:
7491:
7488:
7485:
7482:
7477:
7473:
7469:
7466:
7463:
7460:
7455:
7451:
7430:
7427:
7424:
7419:
7415:
7411:
7406:
7402:
7398:
7395:
7390:
7386:
7365:
7362:
7359:
7354:
7350:
7329:
7326:
7323:
7320:
7296:
7285:
7274:
7271:
7268:
7265:
7245:
7242:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7192:
7188:
7184:
7179:
7175:
7154:
7151:
7148:
7143:
7139:
7127:
7124:
7102:
7080:
7077:
7074:
7071:
7068:
7065:
7062:
7059:
7039:
7036:
7033:
7030:
7027:
7024:
7021:
7018:
7015:
7012:
7009:
6989:
6986:
6983:
6980:
6977:
6974:
6971:
6952:
6947:
6942:
6939:
6913:
6891:
6888:
6860:
6856:
6851: in
6848:
6840:
6837:
6834:
6831:
6828:
6825:
6821:
6801:
6798:
6795:
6792:
6770:
6767:
6747:
6727:
6707:
6704:
6701:
6698:
6674:
6650:
6647:
6644:
6641:
6638:
6635:
6613:
6610:
6607:
6604:
6601:
6598:
6595:
6592:
6587:
6583:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6555:
6552:
6549:
6546:
6541:
6538:
6535:
6531:
6527:
6524:
6521:
6518:
6515:
6512:
6509:
6506:
6503:
6500:
6497:
6494:
6491:
6488:
6485:
6482:
6479:
6476:
6473:
6470:
6467:
6464:
6459:
6455:
6433:
6407:
6404:
6384:
6381:
6378:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6325:
6320:
6316:
6295:
6269:
6254:Main article:
6251:
6248:
6233:
6230:
6170:
6136:
6106:
6050:tensor product
6036:
6008:
5877:
5874:
5841:
5812:
5783:
5780:
5768:, the dual of
5746:
5743:
5706:
5665:
5628:
5606:
5582:
5558:
5477:
5424:Fourier series
5356:representation
5312:
5309:
5285:
5262:
5250:
5227:
5215:
5184:
5172:
5136:linear algebra
5112:Andrew Gleason
5071:
5068:
5065:
5053:
5050:
5011:
5008:
5005:
5002:
4999:
4971:
4968:
4965:
4961:
4958:
4954:
4951:
4948:
4945:
4941:
4937:
4934:
4928:
4925:
4902:
4899:
4896:
4893:
4890:
4871:
4868:
4788:
4785:
4701:
4696:
4691:
4661:quotient group
4598:
4578:rotation group
4548:
4544:
4540:
4537:
4534:
4531:
4528:
4483:
4480:
4407:
4404:
4369:
4365:
4322:
4318:
4293:
4266:
4246:
4226:
4206:
4192:
4191:
4180:
4170:
4156:
4135:
4134:
4123:
4112:
4102:
4056:
4055:
4043:
4040:
4037:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3949:
3937:
3934:
3931:
3911:
3908:
3905:
3902:
3899:
3875:
3872:
3869:
3866:
3846:
3826:
3823:
3818:
3814:
3810:
3805:
3801:
3797:
3794:
3774:
3769:
3765:
3761:
3756:
3752:
3748:
3745:
3742:
3739:
3736:
3731:
3727:
3723:
3720:
3715:
3711:
3707:
3704:
3684:
3679:
3675:
3671:
3666:
3662:
3658:
3655:
3652:
3649:
3644:
3640:
3636:
3633:
3628:
3624:
3620:
3617:
3614:
3599:left-invariant
3575:
3543:
3540:
3503:isolated point
3499:
3498:
3487:
3484:
3481:
3478:
3450:
3447:
3444:
3441:
3421:
3418:
3415:
3404:
3376:{ 1 } :=
3373:
3363:
3346:
3340:
3332:
3326:
3318:
3266:
3263:
3230:
3227:
3099:
3095:
3091:
3088:
3085:
3082:
3077:
3074:
3070:
3065:
3061:
3056:
3053:
3049:
3028:
3025:
3022:
3017:
3014:
3010:
2996:is said to be
2985:
2982:
2979:
2956:
2936:
2914:
2911:
2907:
2903:
2900:
2888:
2885:
2873:
2870:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2788:
2785:
2765:
2745:
2725:
2705:
2702:
2682:
2662:
2657:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2629:
2624:
2605:
2602:
2599:
2596:
2576:
2550:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2425:
2397:
2377:
2374:
2371:
2368:
2348:
2345:
2342:
2322:
2319:
2316:
2313:
2293:
2290:
2287:
2284:
2274:
2264:
2261:
2259:
2256:
2248:Banach algebra
2187:
2183:
2178:
2173:
2150:
2146:
2141:
2136:
2130:
2127:
2124:
2120:
2100:
2088:
2057:
2045:
2027:
2015:
1988:
1976:
1939:
1927:
1895:
1880:, meaning the
1871:-adic integers
1860:
1848:
1817:
1791:
1746:
1718:
1674:
1635:
1606:
1565:
1494:
1464:
1432:
1429:
1383:
1380:
1360:
1359:
1341:
1340:
1336:
1335:
1312:
1273:
1272:
1203:
1202:
1005:
1004:
997:group topology
971:
970:
944:
943:
895:
892:
864:Fourier series
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
9:
6:
4:
3:
2:
13399:
13388:
13385:
13383:
13380:
13378:
13375:
13374:
13372:
13360:
13356:
13352:
13346:
13342:
13338:
13334:
13330:
13326:
13322:
13318:
13316:0-521-23160-4
13312:
13308:
13304:
13300:
13296:
13292:
13288:
13284:
13282:2-88124-133-6
13278:
13274:
13270:
13266:
13262:
13258:
13253:
13249:
13245:
13241:
13235:
13231:
13226:
13223:
13219:
13215:
13211:
13207:
13203:
13199:
13196:
13192:
13188:
13186:0-226-50051-9
13182:
13178:
13174:
13170:
13166:
13163:
13159:
13155:
13149:
13145:
13141:
13137:
13133:
13132:Hewitt, Edwin
13129:
13126:
13122:
13118:
13112:
13108:
13104:
13100:
13096:
13095:Hewitt, Edwin
13092:
13088:
13084:
13080:
13074:
13070:
13065:
13062:
13058:
13054:
13052:0-521-79540-0
13048:
13044:
13040:
13039:
13034:
13030:
13026:
13022:
13018:
13016:0-387-97926-3
13012:
13008:
13004:
13000:
12996:
12993:
12991:0-8493-8490-7
12987:
12983:
12979:
12975:
12971:
12968:
12964:
12960:
12958:3-540-64241-2
12954:
12950:
12946:
12942:
12938:
12935:
12931:
12927:
12923:
12919:
12915:
12911:
12907:
12906:
12901:
12896:
12892:
12888:
12884:
12882:0-387-90839-0
12878:
12874:
12870:
12865:
12861:
12857:
12853:
12847:
12843:
12839:
12835:
12831:
12830:
12819:, p. 12.
12818:
12813:
12806:
12801:
12799:
12797:
12795:
12788:, p. 48.
12787:
12782:
12775:
12770:
12768:
12766:
12758:
12753:
12747:, p. 61.
12746:
12741:
12734:
12729:
12722:
12717:
12710:
12705:
12698:
12693:
12686:
12681:
12674:
12673:Bourbaki 1998
12669:
12662:
12657:
12650:
12645:
12638:
12637:Bourbaki 1998
12633:
12626:
12625:Bourbaki 1998
12621:
12614:
12613:Bourbaki 1998
12609:
12601:
12596:
12592:
12585:
12578:(3): 217β222.
12577:
12573:
12569:
12562:
12555:
12550:
12543:
12542:Bourbaki 1998
12538:
12531:
12530:Bourbaki 1998
12526:
12519:
12514:
12512:
12510:
12508:
12506:
12504:
12502:
12500:
12498:
12496:
12494:
12492:
12490:
12488:
12480:
12476:
12471:
12464:
12459:
12453:, p. 52.
12452:
12447:
12443:
12425:
12417:
12413:
12407:
12403:
12398:
12393:
12389:
12378:
12375:
12369:
12366:
12360:
12357:
12355:
12352:
12346:
12343:
12337:
12334:
12332:
12329:
12326:
12323:
12317:
12314:
12308:
12305:
12303:
12300:
12297:
12294:
12291:
12288:
12285:
12284:Compact group
12282:
12276:
12273:
12270:
12267:
12266:
12257:
12253:
12250:
12246:
12242:
12238:
12232:
12228:
12222:
12218:
12212:
12208:
12199:
12195:
12194:
12193:
12185:
12172:
12169:
12166:
12160:
12154:
12151:
12145:
12139:
12119:
12116:
12113:
12110:
12107:
12087:
12084:
12081:
12078:
12075:
12072:
12052:
12032:
12012:
12009:
11989:
11981:
11965:
11959:
11956:
11953:
11933:
11927:
11924:
11921:
11901:
11898:
11895:
11892:
11872:
11852:
11843:
11842:
11834:
11821:
11818:
11797:
11786:
11783:
11780:
11777:
11774:
11769:
11765:
11760:
11739:
11736:
11728:
11688:
11685:
11682:
11662:
11642:
11634:
11630:
11628:
11607:
11604:
11581:
11578:
11575:
11565:converges in
11552:
11544:
11540:
11536:
11532:
11531:
11529:
11526:becomes is a
11513:
11505:
11500:
11499:
11486:
11483:
11463:
11443:
11435:
11421:
11418:
11410:
11395:
11387:
11372:
11365:
11364:
11363:
11361:
11345:
11316:
11310:
11307:
11304:
11301:
11281:
11258:
11252:
11249:
11246:
11243:
11240:
11237:
11229:
11214:
11191:
11185:
11182:
11159:
11153:
11150:
11142:
11126:
11106:
11098:
11082:
11059:
11053:
11050:
11030:
11007:
11004:
10984:
10978:
10972:
10969:
10966:
10963:
10943:
10937:
10926:
10923:
10907:
10899:
10898:
10885:
10882:
10862:
10859:
10852:to points in
10851:
10835:
10815:
10812:
10806:
10798:
10783:
10763:
10743:
10740:
10720:
10700:
10692:
10678:
10669:Importantly,
10668:
10667:
10666:
10653:
10650:
10630:
10610:
10602:
10586:
10562:
10542:
10534:
10530:
10514:
10507:
10493:
10490:
10470:
10438:
10423:
10403:
10395:
10378:
10375:
10355:
10335:
10332:
10312:
10292:
10284:
10283:
10270:
10267:
10259:
10244:
10221:
10212:
10197:
10196:
10195:
10193:
10177:
10157:
10148:
10135:
10129:
10120:
10090:
10076:
10075:
10062:
10016:
10013:
10010:
10007:
9997:
9987:
9942:
9939:
9936:
9933:
9913:
9893:
9863:
9860:
9857:
9854:
9834:
9787:
9784:
9764:
9756:
9752:
9751:
9736:
9733:
9730:
9727:
9724:
9721:
9701:
9698:
9695:
9668:
9665:
9645:
9642:
9622:
9602:
9594:
9592:
9578:
9554:
9551:
9548:
9545:
9542:
9515:
9512:
9492:
9449:
9446:
9426:
9406:
9398:
9384:
9374:
9337:
9334:
9331:
9328:
9325:
9322:
9299:
9296:
9276:
9260:
9257:
9254:
9251:
9248:
9245:
9235:
9210:
9207:
9204:
9201:
9198:
9195:
9192:
9189:
9183:
9173:
9146:
9143:
9123:
9110:
9096:
9095:
9094:
9092:
9076:
9043:
9030:
9027:
9024:
9021:
9018:
9015:
9007:
8994:
8984:
8970:
8949:
8929:
8926:
8906:
8886:
8866:
8857:
8855:
8854:
8848:
8835:
8830:
8826:
8822:
8819:
8816:
8813:
8793:
8790:
8785:
8781:
8777:
8772:
8768:
8747:
8744:
8739:
8735:
8714:
8711:
8691:
8671:
8651:
8643:
8635:
8632:
8629:
8626:
8620:
8617:
8614:
8606:
8600:
8596:
8592:
8587:
8583:
8578:
8568:
8552:
8530:
8527:
8524:
8519:
8514:
8510:
8506:
8501:
8496:
8492:
8484:
8479:
8466:
8461:
8458:
8455:
8452:
8446:
8443:
8440:
8432:
8426:
8422:
8418:
8413:
8409:
8404:
8399:
8394:
8390:
8386:
8381:
8377:
8368:
8350:
8347:
8344:
8341:
8335:
8332:
8329:
8321:
8315:
8311:
8307:
8302:
8298:
8293:
8288:
8283:
8279:
8275:
8270:
8266:
8257:
8241:
8235:
8232:
8229:
8209:
8206:
8203:
8195:
8177:
8174:
8171:
8168:
8162:
8159:
8156:
8148:
8142:
8138:
8134:
8129:
8125:
8120:
8115:
8110:
8106:
8102:
8097:
8093:
8072:
8067:
8063:
8059:
8056:
8046:
8042:
8038:
8035:
8014:
8008:
8004:
8000:
7995:
7991:
7986:
7982:
7976:
7973:
7970:
7947:
7944:
7941:
7921:
7918:
7896:
7893:
7890:
7885:
7880:
7876:
7872:
7867:
7862:
7858:
7837:
7815:
7812:
7809:
7804:
7799:
7795:
7791:
7786:
7781:
7777:
7767:
7753:
7750:
7742:
7736:
7732:
7707:
7687:
7667:
7659:
7645:
7639:
7636:
7633:
7627:
7622:
7614:
7605:
7602:
7599:
7593:
7587:
7584:
7581:
7570:
7567:
7564:
7560:
7556:
7550:
7547:
7544:
7541:
7538:
7535:
7532:
7524:
7521:
7518:
7510:
7504:
7501:
7498:
7489:
7483:
7475:
7467:
7461:
7453:
7425:
7417:
7409:
7404:
7396:
7388:
7360:
7352:
7327:
7324:
7321:
7318:
7310:
7294:
7286:
7272:
7269:
7266:
7263:
7240:
7237:
7234:
7231:
7225:
7222:
7219:
7210:
7201:
7190:
7182:
7177:
7149:
7141:
7128:
7125:
7122:
7121:
7120:
7118:
7091:A uniformity
7078:
7075:
7072:
7069:
7066:
7063:
7060:
7057:
7037:
7034:
7028:
7025:
7022:
7019:
7016:
7013:
7010:
6987:
6984:
6978:
6975:
6972:
6950:
6940:
6937:
6930:if for every
6929:
6889:
6886:
6877:
6875:
6858:
6854:
6846:
6838:
6835:
6829:
6819:
6799:
6796:
6793:
6790:
6781:
6768:
6765:
6745:
6725:
6702:
6688:
6672:
6664:
6648:
6642:
6639:
6636:
6624:
6605:
6599:
6596:
6590:
6585:
6577:
6568:
6562:
6556:
6553:
6550:
6539:
6536:
6533:
6529:
6525:
6519:
6516:
6513:
6510:
6507:
6504:
6501:
6498:
6495:
6492:
6486:
6483:
6480:
6471:
6465:
6457:
6444:
6431:
6421:
6405:
6402:
6382:
6379:
6376:
6353:
6350:
6347:
6344:
6338:
6335:
6332:
6323:
6318:
6293:
6285:
6280:
6267:
6257:
6256:Uniform space
6247:
6245:
6239:
6229:
6226:
6225:
6218:
6215:SU(2)/{Β±1} β
6211:
6206:
6205:diffeomorphic
6202:
6198:
6194:
6190:
6185:
6157:
6153:
6093:
6085:
6081:
6078:
6061:
6059:
6055:
6051:
6026:algebra over
6025:
5997:
5993:
5984:
5980:
5976:
5970:
5966:
5962:
5957:
5953:
5945:
5937:
5932:
5930:
5921:
5916:
5912:
5904:
5895:
5891:
5883:
5882:homotopy type
5873:
5871:
5867:
5854:
5829:
5778:
5763:
5741:
5727:
5722:
5719:
5692:
5688:
5684:
5679:
5654:
5648:
5644:
5547:
5543:
5539:
5531:
5527:
5523:
5519:
5515:
5510:
5508:
5504:
5499:
5495:
5464:
5456:for integers
5454:
5450:
5444:
5437:
5430:
5425:
5421:
5417:
5413:
5409:
5405:
5404:finite groups
5400:
5398:
5393:
5389:
5383:
5379:
5374:
5366:
5358:
5357:
5351:
5347:
5343:
5338:
5330:
5326:
5318:
5308:
5300:
5274:
5265:
5239:
5230:
5204:
5197:
5193:
5187:
5160:
5156:
5155:compact group
5151:
5149:
5141:
5137:
5129:
5128:real analytic
5121:
5117:
5113:
5105:
5097:
5096:
5091:
5089:
5085:
5069:
5063:
5049:
5046:
5038:
5030:
5026:
5009:
5003:
5000:
4997:
4988:
4986:
4966:
4949:
4946:
4943:
4939:
4935:
4932:
4923:
4900:
4894:
4891:
4888:
4879:
4877:
4867:
4860:
4856:
4834:
4830:
4826:
4822:
4795:
4784:
4782:
4777:
4773:
4756:
4747:
4738:
4730:
4726:
4721:
4718:
4694:
4676:is closed in
4670:
4666:
4662:
4654:
4645:
4640:is closed in
4634:
4630:
4623:
4619:
4615:
4584:
4579:
4574:
4570:
4562:
4546:
4542:
4538:
4532:
4529:
4526:
4514:
4510:
4505:
4501:
4497:
4479:
4474:is normal in
4448:
4444:
4440:
4434:
4417:
4413:
4403:
4393:
4367:
4363:
4346:
4338:
4320:
4316:
4307:
4306:Uffe Haagerup
4283:
4280:
4196:
4181:
4178:
4171:
4168:
4164:
4157:
4154:
4151:
4144:
4143:
4142:
4124:
4113:
4110:
4103:
4096:
4088:
4084:
4077:
4076:
4075:
4069:
4065:
4061:
4041:
4038:
4035:
4012:
4009:
4003:
4000:
3997:
3991:
3988:
3985:
3982:
3979:
3973:
3967:
3961:
3953:
3950:
3935:
3932:
3929:
3906:
3903:
3900:
3889:
3873:
3870:
3864:
3844:
3824:
3821:
3816:
3812:
3808:
3803:
3799:
3795:
3792:
3767:
3763:
3759:
3754:
3750:
3743:
3740:
3734:
3729:
3725:
3721:
3718:
3713:
3709:
3702:
3677:
3673:
3669:
3664:
3660:
3653:
3650:
3642:
3638:
3634:
3631:
3626:
3622:
3618:
3612:
3604:
3600:
3597:
3596:
3595:
3573:
3557:
3542:Metrisability
3539:
3533:
3529:
3520:
3516:
3506:
3504:
3485:
3482:
3479:
3476:
3448:
3445:
3442:
3439:
3419:
3416:
3413:
3405:
3393:
3374:
3368:is closed in
3364:
3361:
3341:
3338:
3327:
3324:
3313:
3312:
3311:
3305:
3300:
3297:
3293:
3262:
3260:
3256:
3252:
3244:
3240:
3236:
3235:uniform space
3229:Uniform space
3226:
3223:
3216:
3212:
3194:
3185:
3182:
3179:
3173:
3169:
3166:
3148:
3145:
3142:
3136:
3132:
3126:
3122:
3097:
3093:
3089:
3086:
3083:
3080:
3075:
3072:
3068:
3063:
3059:
3054:
3051:
3047:
3026:
3023:
3020:
3015:
3012:
3008:
2999:
2983:
2980:
2977:
2968:
2954:
2934:
2912:
2909:
2905:
2898:
2884:
2871:
2868:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2812:
2809:
2806:
2786:
2783:
2763:
2743:
2723:
2703:
2700:
2680:
2650:
2647:
2644:
2641:
2638:
2632:
2622:
2603:
2600:
2597:
2594:
2587:then for all
2574:
2566:
2526:
2520:
2517:
2514:
2511:
2508:
2505:
2499:
2496:
2493:
2470:
2467:
2464:
2461:
2458:
2455:
2449:
2446:
2443:
2423:
2415:
2411:
2395:
2375:
2372:
2369:
2366:
2346:
2343:
2340:
2320:
2317:
2311:
2291:
2288:
2285:
2282:
2268:
2255:
2253:
2249:
2244:
2242:
2238:
2234:
2230:
2226:
2222:
2221:Hilbert space
2218:
2214:
2210:
2205:
2203:
2185:
2181:
2176:
2148:
2144:
2139:
2128:
2125:
2122:
2118:
2109:
2103:
2075:
2073:
2072:-adic numbers
2071:
2066:
2060:
2030:
2004:
1998:
1991:
1965:
1959:
1957:
1952:
1948:
1942:
1915:
1910:
1883:
1882:inverse limit
1876:
1872:
1870:
1863:
1837:is the group
1836:
1835:number theory
1832:
1829:. This is a
1806:
1779:
1777:
1773:
1769:
1765:
1760:
1734:
1707:
1703:
1698:
1694:
1690:
1662:
1656:
1651:
1623:
1595:
1589:
1554:
1549:
1545:
1540:
1532:
1527:
1522:
1518:
1514:
1510:
1483:
1453:
1448:
1446:
1442:
1438:
1428:
1425:
1424:homeomorphism
1421:
1417:
1412:
1410:
1406:
1401:
1397:
1393:
1389:
1382:Homomorphisms
1379:
1377:
1373:
1369:
1368:group objects
1365:
1357:
1356:
1355:
1352:
1350:
1339:Hausdorffness
1338:
1337:
1332:
1328:
1322:
1318:
1313:
1310:
1306:
1302:
1298:
1291:
1287:
1283:
1278:
1277:
1276:
1270:
1269:
1268:
1265:
1261:
1257:
1231:
1217:
1213:
1209:
1200:
1196:
1192:
1185:
1181:
1177:
1173:
1172:
1171:
1168:
1165:
1161:
1157:
1153:
1147:
1143:
1121:
1111:
1107:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1067:
1063:
1059:
1029:
1019:
1015:
1011:
1002:
1001:
1000:
998:
994:
990:
985:
981:
976:
968:
964:
958:
954:
949:
948:
947:
941:
937:
933:
926:
922:
918:
913:
912:
911:
909:
901:
891:
888:
884:
880:
876:
872:
867:
865:
861:
857:
853:
848:
846:
842:
838:
834:
830:
823:
819:
814:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
19:
13336:
13302:
13272:
13256:
13229:
13209:
13172:
13139:
13102:
13068:
13037:
13002:
12977:
12944:
12909:
12903:
12868:
12837:
12812:
12781:
12752:
12745:Edwards 1995
12740:
12733:Hatcher 2001
12728:
12721:Hatcher 2001
12716:
12704:
12692:
12680:
12668:
12656:
12644:
12632:
12620:
12608:
12590:
12584:
12575:
12571:
12561:
12549:
12537:
12525:
12481:, p. 51
12470:
12465:, p. 1.
12458:
12446:
12423:
12415:
12411:
12405:
12401:
12396:
12392:
12240:
12236:
12230:
12226:
12220:
12216:
12210:
12206:
12191:
11979:
11844:
11836:
11835:
11632:
11631:
11626:
11624:
11359:
11358:is called a
11337:
11140:
11096:
10921:
10849:
10796:
10670:
10600:
10599:is called a
10578:
10191:
10190:is called a
10149:
10054:
10053:
10029:a prefilter
9999:
9802:
9570:
9090:
9045:A prefilter
9044:
8986:
8963:
8858:
8851:
8849:
8566:
8565:is called a
8480:
8366:
8255:
8254:denotes the
8193:
8192:denotes the
7911:is a net in
7830:is a net in
7768:
7738:
7116:
6927:
6926:is called a
6878:
6782:
6662:
6625:
6423:
6419:
6369:and for any
6283:
6281:
6259:
6241:
6223:
6209:
6197:Hermann Weyl
6186:
6156:contractible
6083:
6079:
6062:
5995:
5991:
5983:Armand Borel
5975:Hopf algebra
5968:
5964:
5960:
5951:
5933:
5919:
5893:
5879:
5855:
5827:
5723:
5691:unitary dual
5680:
5652:
5646:
5642:
5526:Haar measure
5511:
5500:
5493:
5462:
5452:
5448:
5442:
5435:
5428:
5401:
5396:
5391:
5387:
5381:
5377:
5372:
5364:
5354:
5349:
5345:
5341:
5336:
5329:group action
5324:
5316:
5314:
5298:
5272:
5263:
5237:
5228:
5202:
5191:
5185:
5152:
5093:
5092:
5055:
5047:
5036:
5028:
5024:
4989:
4880:
4873:
4858:
4854:
4835:
4828:
4824:
4820:
4793:
4790:
4775:
4771:
4754:
4745:
4736:
4722:
4716:
4668:
4664:
4646:
4632:
4628:
4621:
4617:
4613:
4582:
4572:
4511:is called a
4503:
4499:
4485:
4446:
4442:
4438:
4432:
4409:
4386:is open and
4278:
4194:
4193:
4175:is properly
4136:
4059:
4057:
3951:
3602:
3598:
3545:
3518:
3514:
3507:
3500:
3391:
3301:
3295:
3291:
3268:
3251:completeness
3242:
3238:
3232:
3221:
3214:
3210:
3186:
3180:
3177:
3171:
3167:
3164:
3149:
3143:
3140:
3134:
3130:
3124:
3120:
2969:
2967:to itself.
2890:
2266:
2245:
2241:gauge groups
2217:Banach space
2215:, such as a
2206:
2101:
2076:
2069:
2058:
2028:
2002:
1989:
1963:
1955:
1940:
1913:
1908:
1875:prime number
1868:
1861:
1780:
1776:Lie algebras
1761:
1705:
1701:
1654:
1593:
1552:
1541:
1530:
1520:
1517:circle group
1452:real numbers
1449:
1434:
1415:
1413:
1408:
1399:
1395:
1388:homomorphism
1387:
1385:
1361:
1353:
1342:
1330:
1326:
1320:
1316:
1308:
1304:
1300:
1296:
1289:
1285:
1281:
1274:
1263:
1259:
1255:
1229:
1215:
1211:
1207:
1204:
1198:
1194:
1190:
1183:
1179:
1175:
1169:
1163:
1159:
1155:
1151:
1145:
1141:
1119:
1109:
1105:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1065:
1061:
1057:
1027:
1017:
1013:
1009:
1006:
996:
992:
983:
979:
972:
966:
962:
956:
952:
945:
939:
935:
931:
924:
920:
916:
899:
897:
868:
849:
832:
826:
818:real numbers
640:
628:
616:
604:
592:
580:
568:
556:
529:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
13206:Zippin, Leo
12697:Mackey 1976
12479:Bredon 1997
12224:defined by
12200:is a group
10106:; that is,
9986:is Cauchy.
8760:such that
8194:product net
6445:is the set
6395:containing
6306:is the set
6193:Γlie Cartan
5088:submanifold
4753:containing
4727:(i.e., the
4659:, then the
3586:. A metric
2388:the subset
2225:loop groups
1995:as well as
1661:linear maps
1416:isomorphism
1154: := {
829:mathematics
530:Topological
369:alternating
13382:Lie groups
13371:Categories
12826:References
11635:: Suppose
10691:is allowed
9714:such that
9535:such that
8983:-small set
8567:Cauchy net
8367:difference
7311:(that is,
7115:is called
6236:See also:
5979:Heinz Hopf
5956:cohomology
5915:loop space
5899:principal
5408:direct sum
5385:, the map
5120:Leo Zippin
5102:that is a
4559:is always
4347:subgroup,
4177:metrisable
4109:metrisable
3785:) for all
3594:is called
3556:metrisable
3432:such that
3323:Kolmogorov
2258:Properties
2077:The group
1947:Cantor set
1764:Lie groups
1733:isometries
1478:Euclidean
1253:such that
1139:such that
1078: := {
1055:such that
975:continuous
879:in physics
875:symmetries
845:continuity
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
13359:840278135
13248:144216834
12982:CRC Press
12934:119755117
12595:CiteSeerX
12439:Citations
12296:Lie group
12167:∈
12152:−
12117:∈
12111:−
12082:∈
11963:→
11931:→
11896:⊆
11787:∈
11775:
11701:and that
11686:⊆
11582:τ
11409:converges
11311:
11253:
11247:⊆
11241:≠
11238:∅
11186:
11154:
11143:point in
11011:∅
11008:≠
10973:
10967:⊆
10925:Hausdorff
10848:converge
10810:∖
10579:A subset
10258:converges
10216:℘
10213:⊆
10150:A subset
10124:℘
10121:⊆
10085:℘
10011:⊆
9943:
9937:∈
9891:→
9858:∈
9725:⊆
9699:∈
9688:and some
9669:∈
9552:⊆
9546:−
9516:∈
9375:−
9338:∈
9326:−
9274:→
9261:∈
9249:−
9211:∈
9193:−
9174:−
9121:→
9111:−
9089:called a
9025:⊆
9019:−
8823:≥
8791:∈
8778:−
8745:∈
8641:→
8633:×
8627:∈
8593:−
8528:∈
8497:∙
8459:×
8453:∈
8419:−
8395:∙
8387:−
8382:∙
8348:×
8342:∈
8284:∙
8271:∙
8239:→
8233:×
8175:×
8169:∈
8111:∙
8103:×
8098:∙
8060:≤
8039:≤
7983:≤
7945:×
7894:∈
7863:∙
7813:∈
7782:∙
7708:τ
7637:×
7619:Δ
7594:×
7568:∈
7561:⋃
7542:∈
7522:∈
7472:Δ
7468:∘
7450:Δ
7414:Δ
7385:Δ
7349:Δ
7319:−
7309:symmetric
7267:∈
7238:∈
7187:Δ
7174:Δ
7138:Δ
7073:∈
7035:∈
6985:∈
6941:∈
6824:Δ
6794:×
6697:Δ
6643:τ
6600:×
6582:Δ
6563:×
6537:∈
6530:⋃
6517:∈
6511:−
6499:×
6493:∈
6454:Δ
6380:⊆
6351:∈
6315:Δ
5782:^
5745:^
5503:character
5140:connected
5067:→
5007:→
4953:→
4947:
4927:~
4898:→
4536:→
4507:with the
4406:Subgroups
4292:⇒
4265:⇒
4245:⇒
4225:⇒
4205:⇒
4097:for 1 in
4083:Hausdorff
3989:∣
3983:∈
3933:∈
3868:↦
3822:∈
3443:≠
3417:∈
3360:Tychonoff
3337:Hausdorff
3087:∈
3073:−
3052:−
3013:−
2998:symmetric
2981:⊆
2970:A subset
2910:−
2902:↦
2843:∈
2831:∈
2651:∈
2598:∈
2518:∈
2468:∈
2370:⊆
2344:∈
2315:→
2286:∈
2246:In every
2126:≥
2119:∏
1831:countable
1524:, or the
1349:Hausdorff
1315:β :
1280:+ :
915:β
:
860:integrals
701:Conformal
589:Euclidean
196:nilpotent
13301:(1981).
13271:(1986).
13208:(1955),
13171:(1976),
13138:(1970),
13101:(1979),
13087:30593138
13035:(2001),
13001:(1997).
12976:(1995),
12943:(1998),
12263:See also
10535:") when
10529:complete
9803:Suppose
7769:Suppose
7256:because
7050:for all
6284:diagonal
5929:H-spaces
5903:-bundles
5522:integral
4985:category
4827:) β cl (
4412:subgroup
3888:isometry
3522:, where
3480:∉
3406:for any
3395:, where
3335:-space (
3321:-space (
2034:, where
1999:such as
1873:, for a
1622:subspace
1588:matrices
1431:Examples
1405:category
1358:Category
1158: :
1149:, where
1086: :
1069:, where
977:. Here
951: :
885:, every
822:addition
696:PoincarΓ©
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
13325:0606198
13291:0201557
13222:0073104
13195:0396826
13162:0262773
13125:0551496
13061:1867354
13025:1700700
12967:1726779
12926:0716262
12891:0705632
12860:2433295
11075:) then
10348:But if
9571:and if
7340:) then
6685:is the
6148:)/SO(2)
5913:to the
5518:measure
5460:(where
4844:and if
4620:+1)/SO(
4611:, with
3695:(resp.
3601:(resp.
3528:closure
3526:is the
3352:⁄
3218:(where
2412:(resp.
1693:compact
1513:abelian
1411:point.
1370:in the
906:, is a
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
13357:
13347:
13323:
13313:
13289:
13279:
13246:
13236:
13220:
13193:
13183:
13160:
13150:
13123:
13113:
13085:
13075:
13059:
13049:
13023:
13013:
12988:
12965:
12955:
12932:
12924:
12889:
12879:
12858:
12848:
12597:
11543:filter
9312:where
9159:where
7441:) and
6195:, and
6075:has a
5954:, the
5724:For a
5538:Type I
5536:is of
5317:action
5146:up to
5142:group
5126:has a
5118:, and
4569:sphere
4567:, the
4496:cosets
4450:. If
4410:Every
4345:clopen
4163:Polish
3952:proper
3886:is an
3386:N β π©
3366:{ 1 }
3345:is a T
3331:is a T
3317:is a T
3138:, and
3039:where
2414:closed
2239:, and
1772:smooth
1689:length
1482:-space
837:groups
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
12930:S2CID
12384:Notes
12132:then
11725:is a
11675:with
11274:then
11141:every
11097:every
10756:then
10693:: If
10527:is a
9876:Then
8196:. If
8085:Then
7934:Make
6217:SO(3)
6201:SU(2)
6160:SL(2,
6126:SL(2,
6122:SO(2)
6096:SL(2,
6090:is a
6086:into
6052:of a
5958:ring
5864:(the
5696:SL(2,
5395:from
5327:is a
5027:/ker(
4857:β© cl
4823:)(cl
4651:is a
4616:= SO(
4351:, of
4195:Note:
4161:is a
4148:is a
4085:and)
3890:from
3191:is a
2799:then
2563:is a
2416:) in
2106:is a
2065:field
1663:from
1620:as a
1526:torus
881:. In
730:Sp(β)
727:SU(β)
140:image
13355:OCLC
13345:ISBN
13311:ISBN
13277:ISBN
13244:OCLC
13234:ISBN
13181:ISBN
13148:ISBN
13111:ISBN
13083:OCLC
13073:ISBN
13047:ISBN
13011:ISBN
12986:ISBN
12953:ISBN
12877:ISBN
12846:ISBN
12422:dom
12397:i.e.
12234:and
11914:and
11865:and
11845:Let
10956:say
10850:only
10797:also
9962:and
9847:and
8942:then
8899:and
7850:and
7733:and
6661:the
6418:the
6282:The
5981:and
5650:for
5520:and
4874:The
4819:(cl
4809:and
4561:open
4515:for
4436:for
4081:is (
4066:and
4058:The
4039:>
4028:for
4010:<
3546:Let
3277:and
3257:and
2756:and
2410:open
2359:and
1708:) β
1583:-by-
1542:The
1450:The
1409:some
1303:) β¦
1262:) β
1241:and
1197:) β¦
1043:and
973:are
938:) β¦
862:and
856:Weil
854:and
852:Haar
839:and
816:The
724:O(β)
713:Loop
532:and
13341:GTM
12914:doi
12910:264
12427:of
12100:if
11978:is
11545:in
11456:in
10922:not
10920:is
10463:on
10285:If
10237:on
10055:on
9906:in
9777:in
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9635:in
9615:of
9439:in
9419:of
9289:in
9136:in
9008:if
8985:or
8859:If
8704:in
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8569:if
8483:net
8256:sum
7307:is
7287:If
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6718:as
6665:on
6422:or
6286:of
6228:).
5917:of
5685:of
5410:of
5371:on
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5331:of
5315:An
5303:as
5271:GL(
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5190:of
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5033:to
4944:ker
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4761:in
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4647:If
4587:in
4585:+1)
4581:SO(
4486:If
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4279:cf.
4107:is
3590:on
3562:on
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3534:of
3508:If
3465:in
3269:If
3220:cl
3209:cl
3199:in
3187:If
3000:if
2693:in
2539:If
2408:is
2219:or
2067:of
2001:GL(
1962:GL(
1912:as
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1731:of
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1258:β
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1249:in
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1167:}.
1135:in
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1047:of
1039:of
1031:in
1025:of
960:,
928:,
827:In
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
13373::
13353:.
13339:.
13321:MR
13319:.
13309:.
13287:MR
13285:.
13259:.
13242:.
13218:MR
13216:,
13204:;
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13146:,
13134:;
13121:MR
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13097:;
13081:.
13057:MR
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13045:,
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13021:MR
13019:.
13009:.
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12980:,
12963:MR
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12885:.
12875:.
12856:MR
12854:.
12844:.
12840:.
12793:^
12764:^
12593:,
12576:28
12574:.
12570:.
12486:^
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12209:β
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11766:cl
11530:.
11332:).
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11250:cl
11227:).
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9567:).
9184::=
8850:A
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8400::=
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6472::=
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6224:RP
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3505:.
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3294:β
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3147:.
3133:βͺ
3128:,
3123:β©
3060::=
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2500::=
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2204:.
2074:.
1759:.
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1539:.
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1267:.
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999:.
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898:A
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615:U(
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11819:C
11798:}
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11740:.
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11464:X
11444:0
11422:.
11419:X
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11320:}
11317:0
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11305:=
11302:S
11282:S
11262:}
11259:0
11256:{
11244:S
11215:S
11195:}
11192:0
11189:{
11163:}
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11157:{
11127:S
11107:S
11083:S
11063:}
11060:0
11057:{
11054:=
11051:S
11031:S
11005:S
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10982:}
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10941:}
10938:0
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10886:.
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10136:.
10133:)
10130:S
10127:(
10116:C
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10088:(
10063:S
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10017:,
10014:X
10008:S
9972:B
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9934:x
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9864:.
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9835:X
9813:B
9788:.
9785:X
9765:0
9737:.
9734:N
9731:+
9728:x
9722:B
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9623:0
9603:N
9579:X
9555:N
9549:B
9543:B
9521:B
9513:B
9493:N
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9450:,
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9427:0
9407:N
9385:.
9380:B
9370:B
9348:}
9343:B
9335:B
9332::
9329:B
9323:B
9320:{
9300:,
9297:X
9277:0
9271:}
9266:B
9258:B
9255::
9252:B
9246:B
9243:{
9221:}
9216:B
9208:C
9205:,
9202:B
9199::
9196:C
9190:B
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9124:0
9116:B
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9077:X
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9031:.
9028:N
9022:B
9016:B
8995:N
8971:N
8950:B
8930:,
8927:0
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8887:X
8867:B
8836:.
8831:0
8827:i
8820:j
8817:,
8814:i
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8621:j
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8615:i
8612:(
8607:)
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8502:=
8493:x
8467:.
8462:J
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8433:)
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8280:y
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8207:=
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7868:=
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7787:=
7778:x
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7668:X
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7640:N
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7631:(
7628:+
7623:X
7615:=
7612:]
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7597:(
7591:)
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7576:[
7571:X
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7557:=
7554:}
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7490:=
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