2857:
By itself, Definition 3 is not quite as useful as the other two definitions as it lacks some of the properties implied by the others. For example, every quotient space of a space satisfying
Definition 1 or Definition 2 is a space of the same kind. But that does not hold for Definition 3.
2973:
section, there is no universally accepted definition in the literature for compactly generated spaces; but
Definitions 1, 2, 3 from that section are some of the more commonly used. In order to express results in a more concise way, this section will make use of the abbreviations
4137:
on a set induced by a family of functions from CG-1 spaces is also CG-1. And the same holds for CG-2. This follows by combining the results above for disjoint unions and quotient spaces, together with the behavior of final topologies under composition of functions.
58:
in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like
4061:
A quotient space of a CG-3 space is not CG-3 in general. In fact, every CG-2 space is a quotient space of a CG-3 space (namely, some locally compact
Hausdorff space); but there are CG-2 spaces that are not CG-3. For a concrete example, the
3690:, which is not compactly generated (as mentioned in the Examples section, it is anticompact and non-discrete). Another example is the Arens space, which is sequential Hausdorff, hence compactly generated. It contains as a subspace the
2912:
need not be a CW-complex. By contrast, the category of simplicial sets had many convenient properties, including being cartesian closed. The history of the study of repairing this situation is given in the article on the
1609:
For coherent spaces, that corresponds to showing that the space is coherent with a subfamily of the family of subspaces. For example, this provides one way to show that locally compact spaces are compactly generated.
4233:
3203:
Compactly generated
Hausdorff spaces include the Hausdorff version of the various classes of spaces mentioned above as CG-1 or CG-2, namely Hausdorff sequential spaces, Hausdorff first countable spaces,
3697:
In a CG-1 space, every closed set is CG-1. The same does not hold for open sets. For instance, as shown in the
Examples section, there are many spaces that are not CG-1, but they are open in their
1560:
5835:
4301:
2726:
1401:
3919:
4880:
4629:
5878:
5638:
5293:
3107:
5734:
5672:
5571:
3965:
950:
501:
5194:
4741:
1607:
850:
466:
5700:
5600:
3626:
1526:
1225:
1178:
1054:
1030:
596:
545:
416:
4909:
4658:
2638:
1817:
223:
105:
4433:
3803:
3438:
3388:
3359:
3133:
2543:
2251:
1970:
The equivalence between conditions (1) and (2) follows from the fact that every inclusion from a subspace is a continuous map; and on the other hand, every continuous map
1722:
1302:
732:
5320:
3684:
3653:
4796:
4394:
4331:
3862:
2682:
2385:
2310:
771:
5914:
5343:
4473:
3757:
2932:
2199:
2000:
1877:
823:
2986:
to denote each of the three definitions unambiguously. This is summarized in the table below (see the
Definitions section for other equivalent conditions for each).
2589:
1768:
1348:
174:
5097:
4994:
2853:
631:
5941:
5487:
4128:
3992:
2778:
2752:
1152:
331:
2078:
2049:
5369:
976:
5777:
5510:
5435:
5392:
5217:
4959:
4050:
3725:
3485:
3156:
2222:
2101:
1900:
1424:
1248:
1201:
1120:
1097:
901:
706:
572:
439:
382:
5985:
5965:
5754:
5530:
5455:
5412:
5257:
5237:
5153:
5129:
5058:
5034:
5014:
4936:
4838:
4818:
4761:
4700:
4680:
4584:
4564:
4541:
4521:
4493:
4023:
3823:
3777:
3545:
3525:
3505:
3458:
3416:
3289:
3269:
3245:
3176:
2609:
2563:
2501:
2436:
2409:
2350:
2330:
2271:
2163:
2131:
2020:
1953:
1922:
1841:
1788:
1742:
1689:
1657:
1580:
1502:
1471:
1447:
1368:
1322:
1268:
1074:
998:
921:
878:
791:
675:
651:
521:
355:
263:
243:
194:
148:
124:
44:
4096:
2818:
1636:
Informally, a space whose topology is determined by its compact subspaces, or equivalently in this case, by all continuous maps from arbitrary compact spaces.
6840:
2958:
2866:
2469:
2450:
article, condition (1) is well-defined, even though the family of continuous maps from arbitrary compact
Hausdorff spaces is not a set but a proper class.
4307:
from the real line with the positive integers identified to a point is sequential. Both spaces are compactly generated
Hausdorff, but their product
3186:. So among the finite spaces, which are all CG-2, the CG-3 spaces are the ones with the discrete topology. Any finite non-discrete space, like the
3038:
Every CG-3 space is CG-2 and every CG-2 space is CG-1. The converse implications do not hold in general, as shown by some of the examples below.
1967:
article, condition (2) is well-defined, even though the family of continuous maps from arbitrary compact spaces is not a set but a proper class.
5994:
These ideas can be generalized to the non-Hausdorff case. This is useful since identification spaces of
Hausdorff spaces need not be Hausdorff.
4145:
of CG-1 spaces is CG-1. The same holds for CG-2. This is also an application of the results above for disjoint unions and quotient spaces.
2869:
can also be defined by pairing the weak
Hausdorff property with Definition 3, which may be easier to state and work with than Definition 2.
4495:
is compactly generated according to one of the definitions in this article. Since compactly generated spaces are defined in terms of a
2942:
These ideas generalize to the non-Hausdorff case; i.e. with a different definition of compactly generated spaces. This is useful since
2939:
is given below. Another suggestion (1964) was to consider the usual Hausdorff spaces but use functions continuous on compact subsets.
6611:
4164:
6810:
2889:, and can be found in General Topology by Kelley, Topology by Dugundji, Rational Homotopy Theory by Félix, Halperin, and Thomas.
3567:
Subspaces of a compactly generated space are not compactly generated in general, even in the Hausdorff case. For example, the
1481:) as part of the definition, while others don't. The definitions in this article will not comprise any such separation axiom.
6786:
6737:
6681:
4351:
is in the sense of condition (3) in the corresponding article, namely each point has a local base of compact neighborhoods.)
1531:
6841:
https://math.stackexchange.com/questions/4646084/unraveling-the-various-definitions-of-k-space-or-compactly-generated-space
4443:
The continuous functions on compactly generated spaces are those that behave well on compact subsets. More precisely, let
5789:
4246:
6651:
6624:
6585:
6564:
2110:
Informally, a space whose topology is determined by all continuous maps from arbitrary compact Hausdorff spaces.
269:
with respect to this family of maps. And other variations of the definition replace compact spaces with compact
5574:
2893:
277:
2643:
Every space satisfying Definition 3 also satisfies Definition 2. The converse is not true. For example, the
2453:
Every space satisfying Definition 2 also satisfies Definition 1. The converse is not true. For example, the
6026:
6337:
2687:
2464:
Definition 2 is the one more commonly used in algebraic topology. This definition is often paired with the
6550:
1203:
for example all compact subspaces, or all compact Hausdorff subspaces. This corresponds to choosing a set
6828:
1373:
394:
in the literature. These definitions share a common structure, starting with a suitably specified family
6720:
4157:
of two compactly generated spaces need not be compactly generated, even if both spaces are Hausdorff and
4058:
space is CG-2. Conversely, every CG-2 space is the quotient space of a locally compact Hausdorff space.
3885:
3864:
is also a quotient map on a locally compact Hausdorff space). The same is true more generally for every
2921:
1613:
Below are some of the more commonly used definitions in more detail, in increasing order of specificity.
1403:
Another choice is to take the family of all continuous maps from arbitrary spaces of a certain type into
4843:
4592:
3015:
Topology same as final topology with respect to continuous maps from arbitrary compact Hausdorff spaces
5846:
5606:
5265:
3698:
3461:
3077:
2454:
6816:
6135:
5705:
5643:
5542:
4543:
with the various maps in the family used to define the final topology. The specifics are as follows.
3924:
2931:
of compactly generated Hausdorff spaces, which is in fact cartesian closed. These ideas extend on the
926:
477:
6855:
6008:
5166:
4708:
4587:
4355:
4055:
4026:
3998:
3880:
3656:
3205:
3056:
2897:
2439:
1925:
1585:
1426:
for example all such maps from arbitrary compact spaces, or from arbitrary compact Hausdorff spaces.
828:
444:
281:
5677:
5577:
3573:
1507:
1206:
1159:
1035:
1011:
577:
526:
397:
6860:
6700:
6017:
4885:
4634:
2614:
1793:
199:
81:
4403:
3782:
3421:
3371:
3342:
3112:
2892:
The motivation for their deeper study came in the 1960s from well known deficiencies of the usual
2522:
2230:
1701:
1624:
is unambiguous and refers to a compactly generated space (in any of the definitions) that is also
1281:
711:
5298:
4703:
3662:
3631:
3507:
is compact, hence CG-1. But it is not CG-2 because open subspaces inherit the CG-2 property and
3031:
For Hausdorff spaces the properties CG-1, CG-2, CG-3 are equivalent. Such spaces can be called
6813:- contains an excellent catalog of properties and constructions with compactly generated spaces
6715:
6669:
6638:
4766:
4373:
4363:
4310:
4002:
3828:
3197:
2649:
2355:
2276:
1956:
737:
5883:
5325:
4446:
3730:
3659:
is compact Hausdorff, hence compactly generated. Its subspace with all limit ordinals except
2172:
1973:
1850:
1484:
As an additional general note, a sufficient condition that can be useful to show that a space
796:
358:
284:
while still containing the typical spaces of interest, which makes them convenient for use in
6003:
5988:
5037:
4500:
4336:
However, in some cases the product of two compactly generated spaces is compactly generated:
3052:
2568:
1747:
1327:
880:
are the closed sets in its k-ification, with a corresponding characterization. In the space
153:
5067:
4964:
2823:
601:
6747:
5919:
5784:
5460:
4101:
3970:
3300:
3213:
2943:
2757:
2731:
1125:
304:
64:
6756:
3418:
that is not CG-1 (for example the Arens-Fort space or an uncountable product of copies of
2728:
does not satisfy Definition 3, because its compact Hausdorff subspaces are the singletons
2054:
2025:
8:
5348:
4397:
2512:
1692:
1275:
955:
5759:
5492:
5417:
5374:
5199:
4941:
4063:
4032:
3707:
3467:
3223:
To provide examples of spaces that are not compactly generated, it is useful to examine
3187:
3138:
2644:
2204:
2083:
1882:
1406:
1230:
1183:
1102:
1079:
883:
688:
554:
421:
364:
6778:
6725:
6673:
5970:
5950:
5840:
5739:
5515:
5440:
5397:
5242:
5222:
5138:
5114:
5043:
5019:
4999:
4921:
4823:
4803:
4746:
4685:
4665:
4569:
4549:
4526:
4506:
4478:
4154:
4008:
3808:
3762:
3530:
3510:
3490:
3443:
3401:
3362:
3336:
3274:
3254:
3230:
3161:
2950:
2936:
2905:
2594:
2548:
2515:
with the family of its compact Hausdorff subspaces; namely, it satisfies the property:
2486:
2421:
2394:
2335:
2315:
2256:
2148:
2116:
2005:
1938:
1907:
1826:
1773:
1727:
1674:
1642:
1565:
1487:
1456:
1432:
1353:
1307:
1253:
1059:
983:
906:
863:
776:
660:
636:
506:
340:
285:
248:
228:
179:
133:
109:
29:
6011: – topological space in which the topology is determined by its countable subsets
5414:
coincide, and the induced topologies on compact subsets are the same. It follows that
4069:
2791:
6792:
6782:
6733:
6711:
6687:
6677:
6647:
6620:
6606:
6581:
6573:
6560:
6210:
4304:
4236:
3391:
3183:
2927:
The first suggestion (1962) to remedy this situation was to restrict oneself to the
2901:
2781:
2480:
Informally, a space whose topology is determined by its compact Hausdorff subspaces.
2461:
is compact and hence satisfies Definition 1, but it does not satisfies Definition 2.
334:
75:
24:
4054:
A quotient space of a CG-2 space is CG-2. In particular, every quotient space of a
3727:
every closed set is CG-2; and so is every open set (because there is a quotient map
276:
Compactly generated spaces were developed to remedy some of the shortcomings of the
6268:
6225:
6211:"Monoidal closed, Cartesian closed and convenient categories of topological spaces"
6067:
5780:
4367:
4366:
of compactly generated spaces (like all CG-1 spaces or all CG-2 spaces), the usual
4158:
4025:
is the quotient space of a weakly locally compact space, which can be taken as the
3691:
3687:
3314:
3307:
3048:
2928:
2458:
6768:
6743:
6729:
6554:
6170:
4344:
4240:
2412:
1929:
1625:
1474:
270:
127:
60:
3398:
For examples of spaces that are CG-1 and not CG-2, one can start with any space
1616:
For Hausdorff spaces, all three definitions are equivalent. So the terminology
5944:
5260:
5132:
5061:
4496:
4134:
3865:
3331:
Other examples of (Hausdorff) spaces that are not compactly generated include:
3042:
2954:
2862:
2465:
2447:
2166:
1964:
1844:
1478:
654:
472:
266:
6849:
6774:
6661:
6273:
6256:
6020: – topology in which the intersection of any family of open sets is open
3568:
3193:
1695:
with the family of its compact subspaces; namely, it satisfies the property:
55:
6691:
3306:
The one-point Lindelöfication of an uncountable discrete space (also called
3227:
spaces, that is, spaces whose compact subspaces are all finite. If a space
6634:
6281:
6229:
3321:
3209:
3060:
2785:
6796:
1076:
is equal to its k-ification; equivalently, if every k-open set is open in
4475:
be a function from a topological space to another and suppose the domain
4340:
The product of two first countable spaces is first countable, hence CG-2.
4066:
is not CG-3, but is homeomorphic to the quotient of the compact interval
225:
Other definitions use a family of continuous maps from compact spaces to
6443:
6071:
5536:
3217:
2909:
6515:
903:
every open set is k-open and every closed set is k-closed. The space
6455:
6356:
6236:
4435:
does belong to the expected category and is the categorical product.
4142:
2784:
instead. On the other hand, it satisfies Definition 2 because it is
78:
with the family of its compact subspaces, meaning that for every set
6479:
6395:
6152:
6150:
3994:
is CG-1. The corresponding statements also hold for CG-2 and CG-3.
6431:
6419:
6344:
6317:
6039:
5100:
4001:
of a CG-1 space is CG-1. In particular, every quotient space of a
3067:
2886:
20:
6407:
6368:
6527:
6293:
6147:
6078:
3025:
Topology coherent with family of its compact Hausdorff subspaces
1180:, one can take all the inclusions maps from certain subspaces of
6503:
3868:
set, that is, the intersection of an open set and a closed set.
3559:
section for the meaning of the abbreviations CG-1, CG-2, CG-3.)
3200:
spaces are CG-1, but not necessarily CG-2 (see examples below).
6596:
6467:
6192:
6190:
3271:
has the discrete topology and the corresponding k-ification of
2051:
and thus factors through the inclusion of the compact subspace
1429:
These different choices for the family of continuous maps into
2908:
is not always an identification map, and the usual product of
2865:
spaces Definitions 2 and 3 are equivalent. Thus the category
6385:
6383:
6058:
Lawson, J.; Madison, B. (1974). "Quotients of k-semigroups".
4228:{\displaystyle X=\mathbb {R} \setminus \{1,1/2,1/3,\ldots \}}
441:
The various definitions differ in their choice of the family
280:. In particular, under some of the definitions, they form a
265:
to be compactly generated if its topology coincides with the
6187:
4396:
is not compactly generated in general, so cannot serve as a
2141:
if it satisfies any of the following equivalent conditions:
1667:
if it satisfies any of the following equivalent conditions:
67:) in the definition of one or both terms, and others don't.
6832:
6820:
6491:
6139:
2914:
6380:
2959:
category CGWH of compactly generated weak Hausdorff spaces
2470:
category CGWH of compactly generated weak Hausdorff spaces
3967:
of topological spaces is CG-1 if and only if each space
296:
6601:, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk
3291:
is the discrete topology. Therefore, any anticompact T
3135:
is the empty set or a single point, which is closed in
3109:
its intersection with every compact Hausdorff subspace
3005:
Topology coherent with family of its compact subspaces
6107:
6105:
1555:{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}
6305:
6117:
6090:
5973:
5953:
5922:
5886:
5849:
5792:
5762:
5742:
5708:
5680:
5646:
5609:
5580:
5545:
5518:
5495:
5463:
5443:
5420:
5400:
5377:
5351:
5328:
5301:
5268:
5245:
5225:
5202:
5169:
5141:
5117:
5070:
5046:
5022:
5002:
4967:
4944:
4924:
4888:
4846:
4826:
4806:
4769:
4749:
4711:
4688:
4668:
4637:
4595:
4572:
4552:
4529:
4509:
4481:
4449:
4406:
4376:
4313:
4249:
4167:
4104:
4072:
4035:
4011:
3973:
3927:
3888:
3831:
3811:
3785:
3765:
3733:
3710:
3665:
3634:
3576:
3533:
3513:
3493:
3470:
3446:
3424:
3404:
3374:
3345:
3277:
3257:
3233:
3164:
3141:
3115:
3080:
2826:
2794:
2780:, and the coherent topology they induce would be the
2760:
2734:
2690:
2652:
2617:
2597:
2571:
2551:
2525:
2489:
2424:
2397:
2358:
2338:
2318:
2279:
2259:
2233:
2207:
2175:
2151:
2119:
2086:
2057:
2028:
2008:
1976:
1941:
1910:
1885:
1853:
1829:
1796:
1776:
1750:
1730:
1704:
1677:
1645:
1588:
1568:
1534:
1510:
1490:
1459:
1435:
1409:
1376:
1356:
1330:
1310:
1284:
1256:
1233:
1209:
1186:
1162:
1128:
1105:
1082:
1062:
1038:
1014:
986:
958:
929:
909:
886:
866:
831:
799:
779:
740:
714:
691:
663:
639:
604:
580:
557:
529:
509:
480:
447:
424:
400:
367:
343:
307:
251:
231:
202:
182:
156:
136:
112:
84:
32:
6022:
Pages displaying wikidata descriptions as a fallback
6013:
Pages displaying wikidata descriptions as a fallback
5674:
with objects the Hausdorff spaces. The functor from
3045:
spaces the properties CG-2 and CG-3 are equivalent.
6102:
5830:{\displaystyle \mathbf {CGTop} \to \mathbf {Top} .}
3295:non-discrete space is not CG-1. Examples include:
677:. The open sets in the k-ification are called the
386:There are multiple (non-equivalent) definitions of
5979:
5959:
5935:
5908:
5872:
5829:
5771:
5748:
5728:
5694:
5666:
5632:
5594:
5565:
5524:
5504:
5481:
5449:
5429:
5406:
5386:
5363:
5337:
5314:
5287:
5251:
5231:
5211:
5188:
5155:that is compactly generated, sometimes called the
5147:
5123:
5091:
5052:
5028:
5008:
4988:
4953:
4930:
4903:
4874:
4832:
4812:
4790:
4755:
4735:
4694:
4674:
4652:
4623:
4578:
4558:
4535:
4515:
4487:
4467:
4427:
4388:
4325:
4295:
4227:
4122:
4090:
4044:
4017:
3986:
3959:
3913:
3856:
3817:
3797:
3771:
3751:
3719:
3678:
3647:
3620:
3539:
3519:
3499:
3479:
3452:
3432:
3410:
3382:
3353:
3283:
3263:
3239:
3170:
3150:
3127:
3101:
2953:, this property is most commonly coupled with the
2877:Compactly generated spaces were originally called
2847:
2812:
2772:
2746:
2720:
2676:
2632:
2603:
2583:
2557:
2537:
2495:
2430:
2403:
2379:
2344:
2324:
2304:
2265:
2245:
2216:
2193:
2169:with respect to the family of all continuous maps
2157:
2125:
2095:
2072:
2043:
2014:
1994:
1947:
1916:
1894:
1871:
1847:with respect to the family of all continuous maps
1835:
1811:
1782:
1762:
1736:
1716:
1683:
1651:
1601:
1574:
1554:
1520:
1496:
1465:
1441:
1418:
1395:
1362:
1342:
1316:
1296:
1262:
1242:
1219:
1195:
1172:
1146:
1114:
1091:
1068:
1048:
1024:
992:
970:
944:
915:
895:
872:
844:
817:
785:
765:
726:
700:
669:
645:
625:
590:
566:
539:
515:
495:
460:
433:
410:
376:
349:
325:
257:
237:
217:
188:
168:
142:
118:
99:
38:
5603:with objects the compactly generated spaces, and
4523:in terms of the continuity of the composition of
6847:
4296:{\displaystyle Y=\mathbb {R} /\{1,2,3,\ldots \}}
3190:, is an example of CG-2 space that is not CG-3.
4743:is continuous for each compact Hausdorff space
1032:) if its topology is determined by all maps in
418:of continuous maps from some compact spaces to
6257:"The total negation of a topological property"
6208:
2788:to the quotient space of the compact interval
361:, that is, the collection of all open sets in
6728:reprint of 1978 ed.). Berlin, New York:
6449:
6389:
6057:
4996:denote the space of all continuous maps from
4840:is continuous if and only if the restriction
4005:space is CG-1. Conversely, every CG-1 space
1278:with that family of subspaces; namely, a set
6710:
6287:
5183:
5170:
4851:
4600:
4290:
4266:
4222:
4182:
3087:
3081:
2767:
2761:
2741:
2735:
2715:
2706:
2700:
2691:
2671:
2659:
2224:In other words, it satisfies the condition:
1156:As for the different choices for the family
5457:was compactly generated to start with then
3871:In a CG-3 space, every closed set is CG-3.
2946:of Hausdorff spaces need not be Hausdorff.
6754:
6521:
6461:
6362:
6242:
6156:
6123:
6045:
4438:
3368:The product of uncountably many copies of
2820:obtained by identifying all the points in
6829:Convenient category of topological spaces
6572:
6437:
6323:
6272:
5371:One can show that the compact subsets of
4882:is continuous for each compact Hausdorff
4257:
4175:
3759:for some locally compact Hausdorff space
3426:
3376:
3347:
657:than (or equal to) the original topology
6646:. Chicago: University of Chicago Press.
6612:Categories for the Working Mathematician
6605:
6594:
6485:
6425:
6350:
6311:
6254:
5196:denote the family of compact subsets of
3220:are also Hausdorff compactly generated.
1504:is compactly generated (with respect to
6766:
6660:
6533:
6509:
6299:
6168:
6096:
6084:
1582:is compactly generated with respect to
6848:
6640:A Concise Course in Algebraic Topology
6162:
2721:{\displaystyle \{\emptyset ,\{1\},X\}}
1453:. Additionally, some authors require
1099:or if every k-closed set is closed in
6817:Compactly generated topological space
6549:
6413:
6401:
6196:
6136:compactly generated topological space
6111:
2611:for every compact Hausdorff subspace
297:General framework for the definitions
6698:
6598:On the foundations of k-group theory
6497:
6473:
6374:
6209:Booth, Peter; Tillotson, J. (1980).
3694:, which is not compactly generated.
1473:to satisfy a separation axiom (like
1396:{\displaystyle K\in {\mathcal {C}}.}
1056:, in the sense that the topology on
6633:
3914:{\displaystyle {\coprod }_{i}X_{i}}
2957:property, so that one works in the
1620:compactly generated Hausdorff space
13:
6804:
6335:
5532:(i.e., there are more open sets).
4875:{\displaystyle f\vert _{K}:K\to Y}
4624:{\displaystyle f\vert _{K}:K\to Y}
4354:The product of a CG-2 space and a
4343:The product of a CG-1 space and a
3327:The "Single ultrafilter topology".
2694:
2332:for every compact Hausdorff space
2201:from all compact Hausdorff spaces
1591:
1547:
1537:
1513:
1385:
1212:
1165:
1041:
1017:
936:
834:
583:
532:
487:
450:
403:
14:
6872:
6619:(2nd ed.). Springer-Verlag.
5873:{\displaystyle \mathbf {CGHaus} }
5633:{\displaystyle \mathbf {CGHaus} }
5288:{\displaystyle A\cap K_{\alpha }}
4702:is continuous if and only if the
4586:is continuous if and only if the
4179:
3158:hence the singleton is closed in
3102:{\displaystyle \{x\}\subseteq X,}
3020:
3010:
3000:
1449:lead to different definitions of
54:if its topology is determined by
6615:. Graduate Texts in Mathematics
5866:
5863:
5860:
5857:
5854:
5851:
5820:
5817:
5814:
5806:
5803:
5800:
5797:
5794:
5729:{\displaystyle \mathbf {CGTop} }
5722:
5719:
5716:
5713:
5710:
5688:
5685:
5682:
5667:{\displaystyle \mathbf {CGTop} }
5660:
5657:
5654:
5651:
5648:
5626:
5623:
5620:
5617:
5614:
5611:
5588:
5585:
5582:
5566:{\displaystyle \mathbf {CGTop} }
5559:
5556:
5553:
5550:
5547:
4913:
3960:{\displaystyle (X_{i})_{i\in I}}
2970:
945:{\displaystyle T_{\mathcal {F}}}
496:{\displaystyle T_{\mathcal {F}}}
6338:"A note about the Arens' space"
6329:
6261:Illinois Journal of Mathematics
6248:
6202:
5189:{\displaystyle \{K_{\alpha }\}}
4736:{\displaystyle f\circ u:K\to Y}
4631:is continuous for each compact
2922:convenient categories of spaces
2475:
2105:
1631:
1602:{\displaystyle {\mathcal {G}}.}
923:together with the new topology
845:{\displaystyle {\mathcal {F}}.}
461:{\displaystyle {\mathcal {F}},}
6218:Pacific Journal of Mathematics
6129:
6051:
5903:
5890:
5810:
5695:{\displaystyle \mathbf {Top} }
5595:{\displaystyle \mathbf {Top} }
5219:We define the new topology on
5106:
5086:
5074:
4983:
4971:
4866:
4779:
4727:
4615:
4459:
4422:
4410:
4117:
4105:
4085:
4073:
3942:
3928:
3851:
3845:
3743:
3621:{\displaystyle \omega _{1}+1=}
3615:
3596:
3339:of uncountably many copies of
2894:category of topological spaces
2839:
2827:
2807:
2795:
2565:exactly when the intersection
2368:
2299:
2293:
2185:
2067:
2061:
2038:
2032:
1986:
1863:
1744:exactly when the intersection
1521:{\displaystyle {\mathcal {F}}}
1324:exactly when the intersection
1220:{\displaystyle {\mathcal {C}}}
1173:{\displaystyle {\mathcal {F}}}
1049:{\displaystyle {\mathcal {F}}}
1025:{\displaystyle {\mathcal {F}}}
809:
760:
754:
617:
605:
591:{\displaystyle {\mathcal {F}}}
540:{\displaystyle {\mathcal {F}}}
411:{\displaystyle {\mathcal {F}}}
320:
308:
291:
278:category of topological spaces
70:In the simplest definition, a
16:Property of topological spaces
1:
6757:"The category of CGWH spaces"
6580:. Heldermann Verlag, Berlin.
6543:
6027:K-space (functional analysis)
4904:{\displaystyle K\subseteq X.}
4653:{\displaystyle K\subseteq X.}
3686:removed is isomorphic to the
3556:
3550:
3208:spaces, etc. In particular,
3033:compactly generated Hausdorff
2872:
2633:{\displaystyle K\subseteq X.}
1812:{\displaystyle K\subseteq X.}
1274:exactly when its topology is
218:{\displaystyle K\subseteq X.}
100:{\displaystyle A\subseteq X,}
6755:Strickland, Neil P. (2009).
6701:"Compactly generated spaces"
5111:Given any topological space
4428:{\displaystyle k(X\times Y)}
4333:is not compactly generated.
4029:of the compact subspaces of
3874:
3798:{\displaystyle U\subseteq X}
3562:
3433:{\displaystyle \mathbb {R} }
3383:{\displaystyle \mathbb {Z} }
3354:{\displaystyle \mathbb {R} }
3251:, every compact subspace of
3128:{\displaystyle K\subseteq X}
2538:{\displaystyle A\subseteq X}
2415:of compact Hausdorff spaces.
2246:{\displaystyle A\subseteq X}
1717:{\displaystyle A\subseteq X}
1297:{\displaystyle A\subseteq X}
1008:(with respect to the family
727:{\displaystyle U\subseteq X}
7:
6721:Counterexamples in Topology
5997:
5437:is compactly generated. If
5315:{\displaystyle K_{\alpha }}
4148:
3679:{\displaystyle \omega _{1}}
3648:{\displaystyle \omega _{1}}
3074:(because given a singleton
2964:
1790:for every compact subspace
574:Since all the functions in
523:with respect to the family
196:for every compact subspace
10:
6877:
6811:Compactly generated spaces
6767:Willard, Stephen (2004) .
6336:Ma, Dan (19 August 2010).
5489:Otherwise the topology on
4161:. For example, the space
3699:one-point compactification
3462:one-point compactification
3216:are compactly generated.
3057:Alexandrov-discrete spaces
2591:is open (resp. closed) in
2545:is open (resp. closed) in
2455:one-point compactification
2312:is open (resp. closed) in
2253:is open (resp. closed) in
1770:is open (resp. closed) in
1724:is open (resp. closed) in
1350:is open (resp. closed) in
1304:is open (resp. closed) in
6450:Lawson & Madison 1974
6390:Lawson & Madison 1974
6009:Countably generated space
5345:Denote this new space by
5131:we can define a possibly
4791:{\displaystyle u:K\to X.}
4389:{\displaystyle X\times Y}
4356:locally compact Hausdorff
4326:{\displaystyle X\times Y}
4056:locally compact Hausdorff
3857:{\displaystyle q^{-1}(U)}
3657:first uncountable ordinal
3206:locally compact Hausdorff
3051:are CG-2. This includes
2898:cartesian closed category
2677:{\displaystyle X=\{0,1\}}
2440:locally compact Hausdorff
2438:is a quotient space of a
2411:is a quotient space of a
2380:{\displaystyle f:K\to X.}
2352:and every continuous map
2305:{\displaystyle f^{-1}(A)}
1955:is a quotient space of a
1528:) is to find a subfamily
1451:compactly generated space
766:{\displaystyle f^{-1}(U)}
388:compactly generated space
282:cartesian closed category
72:compactly generated space
48:compactly generated space
6595:Lamartin, W. F. (1977),
6288:Steen & Seebach 1995
6032:
6018:Finitely generated space
5909:{\displaystyle k(Y^{X})}
5640:the full subcategory of
5573:the full subcategory of
5338:{\displaystyle \alpha .}
4468:{\displaystyle f:X\to Y}
4098:obtained by identifying
3752:{\displaystyle q:Y\to X}
3303:on an uncountable space.
2933:de Vries duality theorem
2881:, after the German word
2194:{\displaystyle f:K\to X}
1995:{\displaystyle f:K\to X}
1879:from all compact spaces
1872:{\displaystyle f:K\to X}
818:{\displaystyle f:K\to X}
6255:Bankston, Paul (1979).
6169:Hatcher, Allen (2001).
5512:is strictly finer than
4918:For topological spaces
4439:Continuity of functions
4400:. But its k-ification
4347:space is CG-1. (Here,
3527:is an open subspace of
2885:. They were studied by
2584:{\displaystyle A\cap K}
1763:{\displaystyle A\cap K}
1343:{\displaystyle A\cap K}
169:{\displaystyle A\cap K}
6716:Seebach, J. Arthur Jr.
6699:Rezk, Charles (2018).
6670:Upper Saddle River, NJ
6556:Topology and Groupoids
6404:, 5.9.1 (Corollary 2).
6290:, Example 114, p. 136.
6274:10.1215/ijm/1256048236
6230:10.2140/pjm.1980.88.35
5981:
5961:
5937:
5910:
5874:
5831:
5773:
5750:
5730:
5696:
5668:
5634:
5596:
5567:
5526:
5506:
5483:
5451:
5431:
5408:
5388:
5365:
5339:
5316:
5289:
5253:
5239:by declaring a subset
5233:
5213:
5190:
5149:
5125:
5093:
5092:{\displaystyle C(X,Y)}
5054:
5030:
5010:
4990:
4989:{\displaystyle C(X,Y)}
4955:
4932:
4905:
4876:
4834:
4820:is CG-3, the function
4814:
4792:
4757:
4737:
4696:
4682:is CG-2, the function
4676:
4654:
4625:
4580:
4566:is CG-1, the function
4560:
4537:
4517:
4499:, one can express the
4489:
4469:
4429:
4390:
4327:
4297:
4239:from the real line is
4229:
4124:
4092:
4046:
4019:
4003:weakly locally compact
3988:
3961:
3915:
3858:
3819:
3799:
3773:
3753:
3721:
3680:
3649:
3622:
3541:
3521:
3501:
3481:
3454:
3434:
3412:
3384:
3355:
3285:
3265:
3241:
3198:weakly locally compact
3172:
3152:
3129:
3103:
3066:Every CG-3 space is a
3053:first countable spaces
2935:. A definition of the
2849:
2848:{\displaystyle (0,1].}
2814:
2774:
2748:
2722:
2678:
2634:
2605:
2585:
2559:
2539:
2497:
2432:
2405:
2381:
2346:
2326:
2306:
2267:
2247:
2218:
2195:
2159:
2127:
2097:
2074:
2045:
2016:
1996:
1957:weakly locally compact
1949:
1918:
1896:
1873:
1837:
1813:
1784:
1764:
1738:
1718:
1685:
1653:
1603:
1576:
1556:
1522:
1498:
1467:
1443:
1420:
1397:
1364:
1344:
1318:
1298:
1264:
1244:
1221:
1197:
1174:
1148:
1116:
1093:
1070:
1050:
1026:
994:
972:
946:
917:
897:
874:
846:
819:
787:
767:
728:
702:
671:
647:
627:
626:{\displaystyle (X,T),}
592:
568:
541:
517:
497:
462:
435:
412:
378:
351:
327:
259:
239:
219:
190:
170:
144:
120:
101:
40:
6377:, Proposition 3.4(3).
6004:Compact-open topology
5989:compact-open topology
5982:
5962:
5938:
5936:{\displaystyle Y^{X}}
5911:
5875:
5832:
5774:
5751:
5731:
5697:
5669:
5635:
5597:
5568:
5535:This construction is
5527:
5507:
5484:
5482:{\displaystyle kX=X.}
5452:
5432:
5409:
5389:
5366:
5340:
5317:
5290:
5254:
5234:
5214:
5191:
5163:of the topology. Let
5150:
5126:
5103:equivalence classes.
5094:
5055:
5038:compact-open topology
5031:
5011:
4991:
4956:
4933:
4906:
4877:
4835:
4815:
4793:
4758:
4738:
4697:
4677:
4655:
4626:
4581:
4561:
4538:
4518:
4490:
4470:
4430:
4391:
4328:
4298:
4230:
4125:
4123:{\displaystyle (0,1]}
4093:
4047:
4020:
3989:
3987:{\displaystyle X_{i}}
3962:
3916:
3859:
3820:
3800:
3774:
3754:
3722:
3681:
3650:
3623:
3542:
3522:
3502:
3482:
3455:
3435:
3413:
3385:
3361:(each with the usual
3356:
3286:
3266:
3242:
3214:topological manifolds
3173:
3153:
3130:
3104:
2944:identification spaces
2896:. This fails to be a
2850:
2815:
2775:
2773:{\displaystyle \{1\}}
2749:
2747:{\displaystyle \{0\}}
2723:
2679:
2635:
2606:
2586:
2560:
2540:
2498:
2468:property to form the
2433:
2406:
2382:
2347:
2327:
2307:
2268:
2248:
2219:
2196:
2160:
2128:
2098:
2075:
2046:
2017:
2002:from a compact space
1997:
1950:
1919:
1897:
1874:
1838:
1814:
1785:
1765:
1739:
1719:
1686:
1654:
1604:
1577:
1557:
1523:
1499:
1468:
1444:
1421:
1398:
1365:
1345:
1319:
1299:
1265:
1245:
1222:
1198:
1175:
1149:
1147:{\displaystyle kX=X.}
1117:
1094:
1071:
1051:
1027:
995:
973:
947:
918:
898:
875:
847:
820:
788:
768:
729:
703:
672:
648:
628:
598:were continuous into
593:
569:
542:
518:
498:
463:
436:
413:
379:
352:
328:
326:{\displaystyle (X,T)}
260:
240:
220:
191:
171:
145:
121:
102:
41:
6416:, Proposition 5.9.1.
5971:
5951:
5920:
5884:
5847:
5790:
5760:
5740:
5706:
5678:
5644:
5607:
5578:
5543:
5516:
5493:
5461:
5441:
5418:
5398:
5375:
5349:
5326:
5299:
5266:
5243:
5223:
5200:
5167:
5139:
5115:
5068:
5044:
5020:
5000:
4965:
4942:
4922:
4886:
4844:
4824:
4804:
4767:
4747:
4709:
4686:
4666:
4635:
4593:
4570:
4550:
4527:
4507:
4479:
4447:
4404:
4374:
4311:
4247:
4165:
4133:More generally, any
4102:
4070:
4033:
4009:
3971:
3925:
3886:
3829:
3809:
3783:
3779:and for an open set
3763:
3731:
3708:
3663:
3632:
3574:
3531:
3511:
3491:
3468:
3444:
3422:
3402:
3372:
3343:
3301:cocountable topology
3275:
3255:
3247:is anticompact and T
3231:
3162:
3139:
3113:
3078:
2969:As explained in the
2824:
2792:
2758:
2732:
2688:
2650:
2615:
2595:
2569:
2549:
2523:
2487:
2483:A topological space
2446:As explained in the
2422:
2395:
2356:
2336:
2316:
2277:
2257:
2231:
2205:
2173:
2149:
2145:(1) The topology on
2117:
2113:A topological space
2084:
2073:{\displaystyle f(K)}
2055:
2044:{\displaystyle f(K)}
2026:
2022:has a compact image
2006:
1974:
1963:As explained in the
1939:
1908:
1883:
1851:
1827:
1823:(2) The topology on
1794:
1774:
1748:
1728:
1702:
1675:
1671:(1) The topology on
1643:
1639:A topological space
1586:
1566:
1532:
1508:
1488:
1457:
1433:
1407:
1374:
1354:
1328:
1308:
1282:
1254:
1231:
1207:
1184:
1160:
1126:
1103:
1080:
1060:
1036:
1012:
984:
956:
927:
907:
884:
864:
829:
797:
777:
738:
712:
689:
661:
637:
602:
578:
555:
527:
507:
478:
445:
422:
398:
365:
341:
305:
249:
229:
200:
180:
154:
134:
110:
82:
65:weak Hausdorff space
30:
6668:(Second ed.).
6524:, Proposition 1.11.
6488:, Proposition 1.11.
5364:{\displaystyle kX.}
5036:topologized by the
4763:and continuous map
4398:categorical product
3805:the restriction of
3035:without ambiguity.
2906:identification maps
2511:if its topology is
2505:compactly-generated
2165:coincides with the
2135:compactly-generated
1843:coincides with the
1661:compactly-generated
1272:compactly generated
1002:compactly generated
971:{\displaystyle kX.}
952:is usually denoted
633:the k-ification of
468:as detailed below.
74:is a space that is
6779:Dover Publications
6712:Steen, Lynn Arthur
6674:Prentice Hall, Inc
6607:Mac Lane, Saunders
6574:Engelking, Ryszard
6476:, Proposition 7.5.
6464:, Proposition 2.6.
6452:, Proposition 1.2.
6428:, Proposition 1.7.
6365:, Proposition 2.2.
6353:, Proposition 1.8.
6245:, Proposition 1.6.
6172:Algebraic Topology
6087:, Definition 43.8.
6072:10.1007/BF02194829
5977:
5957:
5933:
5906:
5870:
5841:exponential object
5827:
5772:{\displaystyle kX}
5769:
5746:
5726:
5692:
5664:
5630:
5592:
5563:
5522:
5505:{\displaystyle kX}
5502:
5479:
5447:
5430:{\displaystyle kX}
5427:
5404:
5387:{\displaystyle kX}
5384:
5361:
5335:
5312:
5285:
5249:
5229:
5212:{\displaystyle X.}
5209:
5186:
5145:
5121:
5099:are precisely the
5089:
5050:
5026:
5006:
4986:
4954:{\displaystyle Y,}
4951:
4928:
4901:
4872:
4830:
4810:
4788:
4753:
4733:
4692:
4672:
4650:
4621:
4576:
4556:
4533:
4513:
4485:
4465:
4425:
4386:
4362:When working in a
4323:
4293:
4225:
4120:
4088:
4045:{\displaystyle X.}
4042:
4015:
3984:
3957:
3911:
3854:
3815:
3795:
3769:
3749:
3720:{\displaystyle X,}
3717:
3676:
3645:
3618:
3547:that is not CG-2.
3537:
3517:
3497:
3480:{\displaystyle Y.}
3477:
3450:
3430:
3408:
3380:
3363:Euclidean topology
3351:
3281:
3261:
3237:
3168:
3151:{\displaystyle K;}
3148:
3125:
3099:
2951:algebraic topology
2937:exponential object
2845:
2810:
2770:
2744:
2718:
2674:
2630:
2601:
2581:
2555:
2535:
2493:
2428:
2401:
2377:
2342:
2322:
2302:
2263:
2243:
2217:{\displaystyle K.}
2214:
2191:
2155:
2123:
2096:{\displaystyle X.}
2093:
2070:
2041:
2012:
1992:
1945:
1932:of compact spaces.
1914:
1895:{\displaystyle K.}
1892:
1869:
1833:
1809:
1780:
1760:
1734:
1714:
1681:
1649:
1599:
1572:
1552:
1518:
1494:
1463:
1439:
1419:{\displaystyle X,}
1416:
1393:
1360:
1340:
1314:
1294:
1260:
1243:{\displaystyle X.}
1240:
1217:
1196:{\displaystyle X,}
1193:
1170:
1144:
1115:{\displaystyle X;}
1112:
1092:{\displaystyle X,}
1089:
1066:
1046:
1022:
990:
968:
942:
913:
896:{\displaystyle X,}
893:
870:
842:
815:
783:
763:
724:
708:they are the sets
701:{\displaystyle X;}
698:
667:
643:
623:
588:
567:{\displaystyle T.}
564:
537:
513:
493:
458:
434:{\displaystyle X.}
431:
408:
377:{\displaystyle X.}
374:
347:
323:
286:algebraic topology
255:
235:
215:
186:
166:
140:
116:
97:
36:
6788:978-0-486-43479-7
6739:978-0-486-68735-3
6683:978-0-13-181629-9
6662:Munkres, James R.
6536:, Problem 43J(1).
6440:, Example 3.3.29.
6326:, Example 1.6.19.
6302:, Problem 43H(2).
6048:, Definition 1.1.
5980:{\displaystyle Y}
5960:{\displaystyle X}
5785:inclusion functor
5749:{\displaystyle X}
5525:{\displaystyle X}
5450:{\displaystyle X}
5407:{\displaystyle X}
5252:{\displaystyle A}
5232:{\displaystyle X}
5148:{\displaystyle X}
5124:{\displaystyle X}
5053:{\displaystyle X}
5029:{\displaystyle Y}
5009:{\displaystyle X}
4931:{\displaystyle X}
4833:{\displaystyle f}
4813:{\displaystyle X}
4756:{\displaystyle K}
4695:{\displaystyle f}
4675:{\displaystyle X}
4579:{\displaystyle f}
4559:{\displaystyle X}
4536:{\displaystyle f}
4516:{\displaystyle f}
4488:{\displaystyle X}
4305:quotient topology
4237:subspace topology
4018:{\displaystyle X}
3818:{\displaystyle q}
3772:{\displaystyle Y}
3701:, which is CG-1.
3540:{\displaystyle X}
3520:{\displaystyle Y}
3500:{\displaystyle X}
3453:{\displaystyle X}
3411:{\displaystyle Y}
3392:discrete topology
3284:{\displaystyle X}
3264:{\displaystyle X}
3240:{\displaystyle X}
3184:discrete topology
3171:{\displaystyle X}
3049:Sequential spaces
3029:
3028:
2902:cartesian product
2782:discrete topology
2604:{\displaystyle K}
2558:{\displaystyle X}
2496:{\displaystyle X}
2431:{\displaystyle X}
2404:{\displaystyle X}
2345:{\displaystyle K}
2325:{\displaystyle K}
2266:{\displaystyle X}
2158:{\displaystyle X}
2126:{\displaystyle X}
2015:{\displaystyle K}
1948:{\displaystyle X}
1917:{\displaystyle X}
1836:{\displaystyle X}
1783:{\displaystyle K}
1737:{\displaystyle X}
1684:{\displaystyle X}
1652:{\displaystyle X}
1575:{\displaystyle X}
1497:{\displaystyle X}
1466:{\displaystyle X}
1442:{\displaystyle X}
1363:{\displaystyle K}
1317:{\displaystyle X}
1263:{\displaystyle X}
1069:{\displaystyle X}
993:{\displaystyle X}
916:{\displaystyle X}
873:{\displaystyle X}
786:{\displaystyle K}
670:{\displaystyle T}
646:{\displaystyle T}
516:{\displaystyle X}
350:{\displaystyle T}
335:topological space
258:{\displaystyle X}
238:{\displaystyle X}
189:{\displaystyle K}
143:{\displaystyle X}
119:{\displaystyle A}
39:{\displaystyle X}
25:topological space
6868:
6856:General topology
6800:
6770:General Topology
6763:
6761:
6751:
6707:
6705:
6695:
6657:
6645:
6630:
6602:
6591:
6578:General Topology
6569:
6537:
6531:
6525:
6519:
6513:
6512:, Theorem 43.10.
6507:
6501:
6495:
6489:
6483:
6477:
6471:
6465:
6459:
6453:
6447:
6441:
6435:
6429:
6423:
6417:
6411:
6405:
6399:
6393:
6387:
6378:
6372:
6366:
6360:
6354:
6348:
6342:
6341:
6333:
6327:
6321:
6315:
6309:
6303:
6297:
6291:
6285:
6279:
6278:
6276:
6252:
6246:
6240:
6234:
6233:
6215:
6206:
6200:
6194:
6185:
6182:See the Appendix
6179:
6177:
6166:
6160:
6154:
6145:
6133:
6127:
6121:
6115:
6109:
6100:
6094:
6088:
6082:
6076:
6075:
6055:
6049:
6043:
6023:
6014:
5986:
5984:
5983:
5978:
5966:
5964:
5963:
5958:
5943:is the space of
5942:
5940:
5939:
5934:
5932:
5931:
5915:
5913:
5912:
5907:
5902:
5901:
5879:
5877:
5876:
5871:
5869:
5836:
5834:
5833:
5828:
5823:
5809:
5778:
5776:
5775:
5770:
5755:
5753:
5752:
5747:
5735:
5733:
5732:
5727:
5725:
5701:
5699:
5698:
5693:
5691:
5673:
5671:
5670:
5665:
5663:
5639:
5637:
5636:
5631:
5629:
5601:
5599:
5598:
5593:
5591:
5572:
5570:
5569:
5564:
5562:
5531:
5529:
5528:
5523:
5511:
5509:
5508:
5503:
5488:
5486:
5485:
5480:
5456:
5454:
5453:
5448:
5436:
5434:
5433:
5428:
5413:
5411:
5410:
5405:
5393:
5391:
5390:
5385:
5370:
5368:
5367:
5362:
5344:
5342:
5341:
5336:
5321:
5319:
5318:
5313:
5311:
5310:
5294:
5292:
5291:
5286:
5284:
5283:
5258:
5256:
5255:
5250:
5238:
5236:
5235:
5230:
5218:
5216:
5215:
5210:
5195:
5193:
5192:
5187:
5182:
5181:
5161:
5160:
5154:
5152:
5151:
5146:
5130:
5128:
5127:
5122:
5098:
5096:
5095:
5090:
5059:
5057:
5056:
5051:
5035:
5033:
5032:
5027:
5015:
5013:
5012:
5007:
4995:
4993:
4992:
4987:
4960:
4958:
4957:
4952:
4937:
4935:
4934:
4929:
4910:
4908:
4907:
4902:
4881:
4879:
4878:
4873:
4859:
4858:
4839:
4837:
4836:
4831:
4819:
4817:
4816:
4811:
4797:
4795:
4794:
4789:
4762:
4760:
4759:
4754:
4742:
4740:
4739:
4734:
4701:
4699:
4698:
4693:
4681:
4679:
4678:
4673:
4659:
4657:
4656:
4651:
4630:
4628:
4627:
4622:
4608:
4607:
4585:
4583:
4582:
4577:
4565:
4563:
4562:
4557:
4542:
4540:
4539:
4534:
4522:
4520:
4519:
4514:
4494:
4492:
4491:
4486:
4474:
4472:
4471:
4466:
4434:
4432:
4431:
4426:
4395:
4393:
4392:
4387:
4368:product topology
4332:
4330:
4329:
4324:
4302:
4300:
4299:
4294:
4265:
4260:
4234:
4232:
4231:
4226:
4212:
4198:
4178:
4129:
4127:
4126:
4121:
4097:
4095:
4094:
4091:{\displaystyle }
4089:
4064:Sierpiński space
4051:
4049:
4048:
4043:
4024:
4022:
4021:
4016:
3993:
3991:
3990:
3985:
3983:
3982:
3966:
3964:
3963:
3958:
3956:
3955:
3940:
3939:
3920:
3918:
3917:
3912:
3910:
3909:
3900:
3899:
3894:
3863:
3861:
3860:
3855:
3844:
3843:
3824:
3822:
3821:
3816:
3804:
3802:
3801:
3796:
3778:
3776:
3775:
3770:
3758:
3756:
3755:
3750:
3726:
3724:
3723:
3718:
3704:In a CG-2 space
3692:Arens-Fort space
3688:Fortissimo space
3685:
3683:
3682:
3677:
3675:
3674:
3654:
3652:
3651:
3646:
3644:
3643:
3627:
3625:
3624:
3619:
3614:
3613:
3586:
3585:
3546:
3544:
3543:
3538:
3526:
3524:
3523:
3518:
3506:
3504:
3503:
3498:
3486:
3484:
3483:
3478:
3459:
3457:
3456:
3451:
3439:
3437:
3436:
3431:
3429:
3417:
3415:
3414:
3409:
3389:
3387:
3386:
3381:
3379:
3360:
3358:
3357:
3352:
3350:
3315:Arens-Fort space
3308:Fortissimo space
3290:
3288:
3287:
3282:
3270:
3268:
3267:
3262:
3246:
3244:
3243:
3238:
3188:Sierpiński space
3182:spaces have the
3177:
3175:
3174:
3169:
3157:
3155:
3154:
3149:
3134:
3132:
3131:
3126:
3108:
3106:
3105:
3100:
2995:Meaning summary
2989:
2988:
2929:full subcategory
2854:
2852:
2851:
2846:
2819:
2817:
2816:
2813:{\displaystyle }
2811:
2779:
2777:
2776:
2771:
2753:
2751:
2750:
2745:
2727:
2725:
2724:
2719:
2683:
2681:
2680:
2675:
2645:Sierpiński space
2639:
2637:
2636:
2631:
2610:
2608:
2607:
2602:
2590:
2588:
2587:
2582:
2564:
2562:
2561:
2556:
2544:
2542:
2541:
2536:
2502:
2500:
2499:
2494:
2459:Arens-Fort space
2437:
2435:
2434:
2429:
2410:
2408:
2407:
2402:
2386:
2384:
2383:
2378:
2351:
2349:
2348:
2343:
2331:
2329:
2328:
2323:
2311:
2309:
2308:
2303:
2292:
2291:
2272:
2270:
2269:
2264:
2252:
2250:
2249:
2244:
2223:
2221:
2220:
2215:
2200:
2198:
2197:
2192:
2164:
2162:
2161:
2156:
2132:
2130:
2129:
2124:
2102:
2100:
2099:
2094:
2079:
2077:
2076:
2071:
2050:
2048:
2047:
2042:
2021:
2019:
2018:
2013:
2001:
1999:
1998:
1993:
1954:
1952:
1951:
1946:
1923:
1921:
1920:
1915:
1901:
1899:
1898:
1893:
1878:
1876:
1875:
1870:
1842:
1840:
1839:
1834:
1818:
1816:
1815:
1810:
1789:
1787:
1786:
1781:
1769:
1767:
1766:
1761:
1743:
1741:
1740:
1735:
1723:
1721:
1720:
1715:
1690:
1688:
1687:
1682:
1658:
1656:
1655:
1650:
1622:
1621:
1608:
1606:
1605:
1600:
1595:
1594:
1581:
1579:
1578:
1573:
1561:
1559:
1558:
1553:
1551:
1550:
1541:
1540:
1527:
1525:
1524:
1519:
1517:
1516:
1503:
1501:
1500:
1495:
1472:
1470:
1469:
1464:
1448:
1446:
1445:
1440:
1425:
1423:
1422:
1417:
1402:
1400:
1399:
1394:
1389:
1388:
1369:
1367:
1366:
1361:
1349:
1347:
1346:
1341:
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1269:
1267:
1266:
1261:
1249:
1247:
1246:
1241:
1227:of subspaces of
1226:
1224:
1223:
1218:
1216:
1215:
1202:
1200:
1199:
1194:
1179:
1177:
1176:
1171:
1169:
1168:
1153:
1151:
1150:
1145:
1122:or in short, if
1121:
1119:
1118:
1113:
1098:
1096:
1095:
1090:
1075:
1073:
1072:
1067:
1055:
1053:
1052:
1047:
1045:
1044:
1031:
1029:
1028:
1023:
1021:
1020:
999:
997:
996:
991:
977:
975:
974:
969:
951:
949:
948:
943:
941:
940:
939:
922:
920:
919:
914:
902:
900:
899:
894:
879:
877:
876:
871:
858:
857:
851:
849:
848:
843:
838:
837:
824:
822:
821:
816:
792:
790:
789:
784:
772:
770:
769:
764:
753:
752:
733:
731:
730:
725:
707:
705:
704:
699:
683:
682:
676:
674:
673:
668:
652:
650:
649:
644:
632:
630:
629:
624:
597:
595:
594:
589:
587:
586:
573:
571:
570:
565:
546:
544:
543:
538:
536:
535:
522:
520:
519:
514:
502:
500:
499:
494:
492:
491:
490:
467:
465:
464:
459:
454:
453:
440:
438:
437:
432:
417:
415:
414:
409:
407:
406:
383:
381:
380:
375:
356:
354:
353:
348:
332:
330:
329:
324:
271:Hausdorff spaces
264:
262:
261:
256:
244:
242:
241:
236:
224:
222:
221:
216:
195:
193:
192:
187:
175:
173:
172:
167:
149:
147:
146:
141:
125:
123:
122:
117:
106:
104:
103:
98:
45:
43:
42:
37:
6876:
6875:
6871:
6870:
6869:
6867:
6866:
6865:
6861:Homotopy theory
6846:
6845:
6807:
6805:Further reading
6789:
6759:
6740:
6730:Springer-Verlag
6703:
6684:
6654:
6643:
6627:
6588:
6567:
6546:
6541:
6540:
6532:
6528:
6522:Strickland 2009
6520:
6516:
6508:
6504:
6496:
6492:
6484:
6480:
6472:
6468:
6462:Strickland 2009
6460:
6456:
6448:
6444:
6436:
6432:
6424:
6420:
6412:
6408:
6400:
6396:
6388:
6381:
6373:
6369:
6363:Strickland 2009
6361:
6357:
6349:
6345:
6334:
6330:
6322:
6318:
6310:
6306:
6298:
6294:
6286:
6282:
6253:
6249:
6243:Strickland 2009
6241:
6237:
6213:
6207:
6203:
6195:
6188:
6175:
6167:
6163:
6159:, Lemma 1.4(c).
6157:Strickland 2009
6155:
6148:
6134:
6130:
6124:Strickland 2009
6122:
6118:
6110:
6103:
6095:
6091:
6083:
6079:
6060:Semigroup Forum
6056:
6052:
6046:Strickland 2009
6044:
6040:
6035:
6021:
6012:
6000:
5972:
5969:
5968:
5952:
5949:
5948:
5945:continuous maps
5927:
5923:
5921:
5918:
5917:
5897:
5893:
5885:
5882:
5881:
5850:
5848:
5845:
5844:
5813:
5793:
5791:
5788:
5787:
5761:
5758:
5757:
5741:
5738:
5737:
5709:
5707:
5704:
5703:
5681:
5679:
5676:
5675:
5647:
5645:
5642:
5641:
5610:
5608:
5605:
5604:
5581:
5579:
5576:
5575:
5546:
5544:
5541:
5540:
5517:
5514:
5513:
5494:
5491:
5490:
5462:
5459:
5458:
5442:
5439:
5438:
5419:
5416:
5415:
5399:
5396:
5395:
5376:
5373:
5372:
5350:
5347:
5346:
5327:
5324:
5323:
5322:for each index
5306:
5302:
5300:
5297:
5296:
5279:
5275:
5267:
5264:
5263:
5244:
5241:
5240:
5224:
5221:
5220:
5201:
5198:
5197:
5177:
5173:
5168:
5165:
5164:
5158:
5157:
5140:
5137:
5136:
5116:
5113:
5112:
5109:
5069:
5066:
5065:
5062:path components
5045:
5042:
5041:
5021:
5018:
5017:
5001:
4998:
4997:
4966:
4963:
4962:
4943:
4940:
4939:
4923:
4920:
4919:
4916:
4887:
4884:
4883:
4854:
4850:
4845:
4842:
4841:
4825:
4822:
4821:
4805:
4802:
4801:
4768:
4765:
4764:
4748:
4745:
4744:
4710:
4707:
4706:
4687:
4684:
4683:
4667:
4664:
4663:
4636:
4633:
4632:
4603:
4599:
4594:
4591:
4590:
4571:
4568:
4567:
4551:
4548:
4547:
4528:
4525:
4524:
4508:
4505:
4504:
4480:
4477:
4476:
4448:
4445:
4444:
4441:
4405:
4402:
4401:
4375:
4372:
4371:
4349:locally compact
4345:locally compact
4312:
4309:
4308:
4261:
4256:
4248:
4245:
4244:
4241:first countable
4208:
4194:
4174:
4166:
4163:
4162:
4151:
4103:
4100:
4099:
4071:
4068:
4067:
4034:
4031:
4030:
4010:
4007:
4006:
3978:
3974:
3972:
3969:
3968:
3945:
3941:
3935:
3931:
3926:
3923:
3922:
3905:
3901:
3895:
3890:
3889:
3887:
3884:
3883:
3877:
3836:
3832:
3830:
3827:
3826:
3810:
3807:
3806:
3784:
3781:
3780:
3764:
3761:
3760:
3732:
3729:
3728:
3709:
3706:
3705:
3670:
3666:
3664:
3661:
3660:
3639:
3635:
3633:
3630:
3629:
3609:
3605:
3581:
3577:
3575:
3572:
3571:
3565:
3553:
3532:
3529:
3528:
3512:
3509:
3508:
3492:
3489:
3488:
3469:
3466:
3465:
3445:
3442:
3441:
3425:
3423:
3420:
3419:
3403:
3400:
3399:
3390:(each with the
3375:
3373:
3370:
3369:
3346:
3344:
3341:
3340:
3294:
3276:
3273:
3272:
3256:
3253:
3252:
3250:
3232:
3229:
3228:
3181:
3163:
3160:
3159:
3140:
3137:
3136:
3114:
3111:
3110:
3079:
3076:
3075:
3071:
2967:
2875:
2825:
2822:
2821:
2793:
2790:
2789:
2759:
2756:
2755:
2733:
2730:
2729:
2689:
2686:
2685:
2651:
2648:
2647:
2616:
2613:
2612:
2596:
2593:
2592:
2570:
2567:
2566:
2550:
2547:
2546:
2524:
2521:
2520:
2488:
2485:
2484:
2478:
2423:
2420:
2419:
2413:topological sum
2396:
2393:
2392:
2357:
2354:
2353:
2337:
2334:
2333:
2317:
2314:
2313:
2284:
2280:
2278:
2275:
2274:
2258:
2255:
2254:
2232:
2229:
2228:
2206:
2203:
2202:
2174:
2171:
2170:
2150:
2147:
2146:
2118:
2115:
2114:
2108:
2085:
2082:
2081:
2056:
2053:
2052:
2027:
2024:
2023:
2007:
2004:
2003:
1975:
1972:
1971:
1940:
1937:
1936:
1930:topological sum
1909:
1906:
1905:
1884:
1881:
1880:
1852:
1849:
1848:
1828:
1825:
1824:
1795:
1792:
1791:
1775:
1772:
1771:
1749:
1746:
1745:
1729:
1726:
1725:
1703:
1700:
1699:
1676:
1673:
1672:
1644:
1641:
1640:
1634:
1619:
1618:
1590:
1589:
1587:
1584:
1583:
1567:
1564:
1563:
1546:
1545:
1536:
1535:
1533:
1530:
1529:
1512:
1511:
1509:
1506:
1505:
1489:
1486:
1485:
1458:
1455:
1454:
1434:
1431:
1430:
1408:
1405:
1404:
1384:
1383:
1375:
1372:
1371:
1355:
1352:
1351:
1329:
1326:
1325:
1309:
1306:
1305:
1283:
1280:
1279:
1255:
1252:
1251:
1232:
1229:
1228:
1211:
1210:
1208:
1205:
1204:
1185:
1182:
1181:
1164:
1163:
1161:
1158:
1157:
1127:
1124:
1123:
1104:
1101:
1100:
1081:
1078:
1077:
1061:
1058:
1057:
1040:
1039:
1037:
1034:
1033:
1016:
1015:
1013:
1010:
1009:
985:
982:
981:
957:
954:
953:
935:
934:
930:
928:
925:
924:
908:
905:
904:
885:
882:
881:
865:
862:
861:
855:
854:
852:Similarly, the
833:
832:
830:
827:
826:
798:
795:
794:
778:
775:
774:
745:
741:
739:
736:
735:
713:
710:
709:
690:
687:
686:
680:
679:
662:
659:
658:
638:
635:
634:
603:
600:
599:
582:
581:
579:
576:
575:
556:
553:
552:
531:
530:
528:
525:
524:
508:
505:
504:
486:
485:
481:
479:
476:
475:
449:
448:
446:
443:
442:
423:
420:
419:
402:
401:
399:
396:
395:
366:
363:
362:
342:
339:
338:
306:
303:
302:
299:
294:
250:
247:
246:
230:
227:
226:
201:
198:
197:
181:
178:
177:
155:
152:
151:
150:if and only if
135:
132:
131:
111:
108:
107:
83:
80:
79:
61:Hausdorff space
31:
28:
27:
17:
12:
11:
5:
6874:
6864:
6863:
6858:
6844:
6843:
6838:
6826:
6814:
6806:
6803:
6802:
6801:
6787:
6764:
6752:
6738:
6708:
6696:
6682:
6658:
6652:
6631:
6625:
6603:
6592:
6586:
6570:
6565:
6545:
6542:
6539:
6538:
6526:
6514:
6502:
6500:, section 3.5.
6490:
6478:
6466:
6454:
6442:
6438:Engelking 1989
6430:
6418:
6406:
6394:
6379:
6367:
6355:
6343:
6328:
6324:Engelking 1989
6316:
6304:
6292:
6280:
6267:(2): 241–252.
6247:
6235:
6201:
6199:, section 5.9.
6186:
6161:
6146:
6128:
6116:
6114:, p. 182.
6101:
6099:, p. 283.
6089:
6077:
6050:
6037:
6036:
6034:
6031:
6030:
6029:
6024:
6015:
6006:
5999:
5996:
5976:
5956:
5930:
5926:
5905:
5900:
5896:
5892:
5889:
5868:
5865:
5862:
5859:
5856:
5853:
5826:
5822:
5819:
5816:
5812:
5808:
5805:
5802:
5799:
5796:
5768:
5765:
5745:
5724:
5721:
5718:
5715:
5712:
5690:
5687:
5684:
5662:
5659:
5656:
5653:
5650:
5628:
5625:
5622:
5619:
5616:
5613:
5590:
5587:
5584:
5561:
5558:
5555:
5552:
5549:
5521:
5501:
5498:
5478:
5475:
5472:
5469:
5466:
5446:
5426:
5423:
5403:
5383:
5380:
5360:
5357:
5354:
5334:
5331:
5309:
5305:
5282:
5278:
5274:
5271:
5261:if and only if
5248:
5228:
5208:
5205:
5185:
5180:
5176:
5172:
5144:
5133:finer topology
5120:
5108:
5105:
5088:
5085:
5082:
5079:
5076:
5073:
5049:
5025:
5005:
4985:
4982:
4979:
4976:
4973:
4970:
4950:
4947:
4927:
4915:
4912:
4900:
4897:
4894:
4891:
4871:
4868:
4865:
4862:
4857:
4853:
4849:
4829:
4809:
4787:
4784:
4781:
4778:
4775:
4772:
4752:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4691:
4671:
4649:
4646:
4643:
4640:
4620:
4617:
4614:
4611:
4606:
4602:
4598:
4575:
4555:
4532:
4512:
4497:final topology
4484:
4464:
4461:
4458:
4455:
4452:
4440:
4437:
4424:
4421:
4418:
4415:
4412:
4409:
4385:
4382:
4379:
4360:
4359:
4358:space is CG-2.
4352:
4341:
4322:
4319:
4316:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4264:
4259:
4255:
4252:
4224:
4221:
4218:
4215:
4211:
4207:
4204:
4201:
4197:
4193:
4190:
4187:
4184:
4181:
4177:
4173:
4170:
4150:
4147:
4135:final topology
4119:
4116:
4113:
4110:
4107:
4087:
4084:
4081:
4078:
4075:
4041:
4038:
4027:disjoint union
4014:
3999:quotient space
3981:
3977:
3954:
3951:
3948:
3944:
3938:
3934:
3930:
3908:
3904:
3898:
3893:
3881:disjoint union
3876:
3873:
3866:locally closed
3853:
3850:
3847:
3842:
3839:
3835:
3814:
3794:
3791:
3788:
3768:
3748:
3745:
3742:
3739:
3736:
3716:
3713:
3673:
3669:
3642:
3638:
3617:
3612:
3608:
3604:
3601:
3598:
3595:
3592:
3589:
3584:
3580:
3564:
3561:
3552:
3549:
3536:
3516:
3496:
3476:
3473:
3449:
3428:
3407:
3396:
3395:
3378:
3366:
3349:
3329:
3328:
3325:
3318:
3311:
3304:
3292:
3280:
3260:
3248:
3236:
3194:Compact spaces
3179:
3167:
3147:
3144:
3124:
3121:
3118:
3098:
3095:
3092:
3089:
3086:
3083:
3069:
3043:weak Hausdorff
3027:
3026:
3023:
3017:
3016:
3013:
3007:
3006:
3003:
2997:
2996:
2993:
2966:
2963:
2955:weak Hausdorff
2949:In modern-day
2874:
2871:
2863:weak Hausdorff
2844:
2841:
2838:
2835:
2832:
2829:
2809:
2806:
2803:
2800:
2797:
2769:
2766:
2763:
2743:
2740:
2737:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2684:with topology
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2641:
2640:
2629:
2626:
2623:
2620:
2600:
2580:
2577:
2574:
2554:
2534:
2531:
2528:
2492:
2477:
2474:
2466:weak Hausdorff
2448:final topology
2444:
2443:
2427:
2416:
2400:
2389:
2388:
2387:
2376:
2373:
2370:
2367:
2364:
2361:
2341:
2321:
2301:
2298:
2295:
2290:
2287:
2283:
2262:
2242:
2239:
2236:
2213:
2210:
2190:
2187:
2184:
2181:
2178:
2167:final topology
2154:
2122:
2107:
2104:
2092:
2089:
2069:
2066:
2063:
2060:
2040:
2037:
2034:
2031:
2011:
1991:
1988:
1985:
1982:
1979:
1965:final topology
1961:
1960:
1944:
1933:
1926:quotient space
1913:
1902:
1891:
1888:
1868:
1865:
1862:
1859:
1856:
1845:final topology
1832:
1821:
1820:
1819:
1808:
1805:
1802:
1799:
1779:
1759:
1756:
1753:
1733:
1713:
1710:
1707:
1680:
1648:
1633:
1630:
1598:
1593:
1571:
1549:
1544:
1539:
1515:
1493:
1479:weak Hausdorff
1462:
1438:
1415:
1412:
1392:
1387:
1382:
1379:
1359:
1339:
1336:
1333:
1313:
1293:
1290:
1287:
1259:
1239:
1236:
1214:
1192:
1189:
1167:
1143:
1140:
1137:
1134:
1131:
1111:
1108:
1088:
1085:
1065:
1043:
1019:
989:
967:
964:
961:
938:
933:
912:
892:
889:
869:
841:
836:
814:
811:
808:
805:
802:
782:
762:
759:
756:
751:
748:
744:
723:
720:
717:
697:
694:
666:
642:
622:
619:
616:
613:
610:
607:
585:
563:
560:
547:is called the
534:
512:
489:
484:
473:final topology
457:
452:
430:
427:
405:
373:
370:
346:
322:
319:
316:
313:
310:
298:
295:
293:
290:
267:final topology
254:
234:
214:
211:
208:
205:
185:
165:
162:
159:
139:
115:
96:
93:
90:
87:
56:compact spaces
35:
15:
9:
6:
4:
3:
2:
6873:
6862:
6859:
6857:
6854:
6853:
6851:
6842:
6839:
6837:
6835:
6830:
6827:
6825:
6823:
6818:
6815:
6812:
6809:
6808:
6798:
6794:
6790:
6784:
6780:
6776:
6775:Mineola, N.Y.
6772:
6771:
6765:
6758:
6753:
6749:
6745:
6741:
6735:
6731:
6727:
6723:
6722:
6717:
6713:
6709:
6702:
6697:
6693:
6689:
6685:
6679:
6675:
6671:
6667:
6663:
6659:
6655:
6653:0-226-51183-9
6649:
6642:
6641:
6636:
6635:May, J. Peter
6632:
6628:
6626:0-387-98403-8
6622:
6618:
6614:
6613:
6608:
6604:
6600:
6599:
6593:
6589:
6587:3-88538-006-4
6583:
6579:
6575:
6571:
6568:
6566:1-4196-2722-8
6562:
6559:, Booksurge,
6558:
6557:
6552:
6551:Brown, Ronald
6548:
6547:
6535:
6530:
6523:
6518:
6511:
6506:
6499:
6494:
6487:
6486:Lamartin 1977
6482:
6475:
6470:
6463:
6458:
6451:
6446:
6439:
6434:
6427:
6426:Lamartin 1977
6422:
6415:
6410:
6403:
6398:
6391:
6386:
6384:
6376:
6371:
6364:
6359:
6352:
6351:Lamartin 1977
6347:
6339:
6332:
6325:
6320:
6313:
6312:Lamartin 1977
6308:
6301:
6296:
6289:
6284:
6275:
6270:
6266:
6262:
6258:
6251:
6244:
6239:
6231:
6227:
6223:
6219:
6212:
6205:
6198:
6193:
6191:
6183:
6174:
6173:
6165:
6158:
6153:
6151:
6144:
6142:
6137:
6132:
6125:
6120:
6113:
6108:
6106:
6098:
6093:
6086:
6081:
6073:
6069:
6065:
6061:
6054:
6047:
6042:
6038:
6028:
6025:
6019:
6016:
6010:
6007:
6005:
6002:
6001:
5995:
5992:
5990:
5974:
5954:
5946:
5928:
5924:
5898:
5894:
5887:
5842:
5837:
5824:
5786:
5782:
5781:right adjoint
5766:
5763:
5743:
5602:
5539:. We denote
5538:
5533:
5519:
5499:
5496:
5476:
5473:
5470:
5467:
5464:
5444:
5424:
5421:
5401:
5381:
5378:
5358:
5355:
5352:
5332:
5329:
5307:
5303:
5295:is closed in
5280:
5276:
5272:
5269:
5262:
5259:to be closed
5246:
5226:
5206:
5203:
5178:
5174:
5162:
5142:
5134:
5118:
5104:
5102:
5083:
5080:
5077:
5071:
5063:
5060:is CG-1, the
5047:
5039:
5023:
5003:
4980:
4977:
4974:
4968:
4948:
4945:
4925:
4914:Miscellaneous
4911:
4898:
4895:
4892:
4889:
4869:
4863:
4860:
4855:
4847:
4827:
4807:
4798:
4785:
4782:
4776:
4773:
4770:
4750:
4730:
4724:
4721:
4718:
4715:
4712:
4705:
4689:
4669:
4660:
4647:
4644:
4641:
4638:
4618:
4612:
4609:
4604:
4596:
4589:
4573:
4553:
4544:
4530:
4510:
4502:
4498:
4482:
4462:
4456:
4453:
4450:
4436:
4419:
4416:
4413:
4407:
4399:
4383:
4380:
4377:
4369:
4365:
4357:
4353:
4350:
4346:
4342:
4339:
4338:
4337:
4334:
4320:
4317:
4314:
4306:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4262:
4253:
4250:
4242:
4238:
4219:
4216:
4213:
4209:
4205:
4202:
4199:
4195:
4191:
4188:
4185:
4171:
4168:
4160:
4156:
4146:
4144:
4139:
4136:
4131:
4114:
4111:
4108:
4082:
4079:
4076:
4065:
4059:
4057:
4052:
4039:
4036:
4028:
4012:
4004:
4000:
3995:
3979:
3975:
3952:
3949:
3946:
3936:
3932:
3906:
3902:
3896:
3891:
3882:
3872:
3869:
3867:
3848:
3840:
3837:
3833:
3812:
3792:
3789:
3786:
3766:
3746:
3740:
3737:
3734:
3714:
3711:
3702:
3700:
3695:
3693:
3689:
3671:
3667:
3658:
3640:
3636:
3610:
3606:
3602:
3599:
3593:
3590:
3587:
3582:
3578:
3570:
3569:ordinal space
3560:
3558:
3548:
3534:
3514:
3494:
3474:
3471:
3463:
3447:
3405:
3393:
3367:
3364:
3338:
3334:
3333:
3332:
3326:
3323:
3319:
3316:
3312:
3309:
3305:
3302:
3298:
3297:
3296:
3278:
3258:
3234:
3226:
3221:
3219:
3215:
3211:
3210:metric spaces
3207:
3201:
3199:
3195:
3191:
3189:
3185:
3165:
3145:
3142:
3122:
3119:
3116:
3096:
3093:
3090:
3084:
3073:
3064:
3062:
3061:finite spaces
3058:
3054:
3050:
3046:
3044:
3039:
3036:
3034:
3024:
3022:
3019:
3018:
3014:
3012:
3009:
3008:
3004:
3002:
2999:
2998:
2994:
2991:
2990:
2987:
2985:
2981:
2977:
2972:
2962:
2960:
2956:
2952:
2947:
2945:
2940:
2938:
2934:
2930:
2925:
2923:
2919:
2917:
2911:
2907:
2903:
2899:
2895:
2890:
2888:
2884:
2880:
2870:
2868:
2864:
2861:However, for
2859:
2855:
2842:
2836:
2833:
2830:
2804:
2801:
2798:
2787:
2783:
2764:
2738:
2712:
2709:
2703:
2697:
2668:
2665:
2662:
2656:
2653:
2646:
2627:
2624:
2621:
2618:
2598:
2578:
2575:
2572:
2552:
2532:
2529:
2526:
2518:
2517:
2516:
2514:
2510:
2506:
2490:
2481:
2473:
2471:
2467:
2462:
2460:
2456:
2451:
2449:
2441:
2425:
2417:
2414:
2398:
2390:
2374:
2371:
2365:
2362:
2359:
2339:
2319:
2296:
2288:
2285:
2281:
2273:exactly when
2260:
2240:
2237:
2234:
2226:
2225:
2211:
2208:
2188:
2182:
2179:
2176:
2168:
2152:
2144:
2143:
2142:
2140:
2136:
2120:
2111:
2106:Definition 2
2103:
2090:
2087:
2064:
2058:
2035:
2029:
2009:
1989:
1983:
1980:
1977:
1968:
1966:
1958:
1942:
1934:
1931:
1927:
1911:
1903:
1889:
1886:
1866:
1860:
1857:
1854:
1846:
1830:
1822:
1806:
1803:
1800:
1797:
1777:
1757:
1754:
1751:
1731:
1711:
1708:
1705:
1697:
1696:
1694:
1678:
1670:
1669:
1668:
1666:
1662:
1646:
1637:
1629:
1627:
1623:
1614:
1611:
1596:
1569:
1542:
1491:
1482:
1480:
1476:
1460:
1452:
1436:
1427:
1413:
1410:
1390:
1380:
1377:
1357:
1337:
1334:
1331:
1311:
1291:
1288:
1285:
1277:
1273:
1257:
1237:
1234:
1190:
1187:
1154:
1141:
1138:
1135:
1132:
1129:
1109:
1106:
1086:
1083:
1063:
1007:
1003:
987:
978:
965:
962:
959:
931:
910:
890:
887:
867:
859:
856:k-closed sets
839:
812:
806:
803:
800:
780:
757:
749:
746:
742:
721:
718:
715:
695:
692:
684:
664:
656:
640:
620:
614:
611:
608:
561:
558:
550:
510:
482:
474:
469:
455:
428:
425:
393:
389:
384:
371:
368:
360:
344:
336:
317:
314:
311:
289:
287:
283:
279:
274:
272:
268:
252:
232:
212:
209:
206:
203:
183:
163:
160:
157:
137:
129:
113:
94:
91:
88:
85:
77:
73:
68:
66:
62:
57:
53:
49:
33:
26:
22:
6833:
6821:
6769:
6719:
6665:
6639:
6616:
6610:
6597:
6577:
6555:
6534:Willard 2004
6529:
6517:
6510:Willard 2004
6505:
6493:
6481:
6469:
6457:
6445:
6433:
6421:
6409:
6397:
6392:, p. 3.
6370:
6358:
6346:
6331:
6319:
6314:, p. 8.
6307:
6300:Willard 2004
6295:
6283:
6264:
6260:
6250:
6238:
6224:(1): 35–53.
6221:
6217:
6204:
6181:
6171:
6164:
6140:
6131:
6119:
6097:Munkres 2000
6092:
6085:Willard 2004
6080:
6063:
6059:
6053:
6041:
5993:
5880:is given by
5838:
5534:
5156:
5110:
4917:
4799:
4661:
4545:
4442:
4361:
4348:
4335:
4243:; the space
4152:
4140:
4132:
4130:to a point.
4060:
4053:
3996:
3921:of a family
3878:
3870:
3703:
3696:
3566:
3554:
3397:
3330:
3322:Appert space
3224:
3222:
3218:CW complexes
3202:
3192:
3178:). Finite T
3065:
3047:
3040:
3037:
3032:
3030:
2992:Abbreviation
2983:
2979:
2975:
2968:
2948:
2941:
2926:
2915:
2910:CW-complexes
2900:, the usual
2891:
2882:
2878:
2876:
2860:
2856:
2786:homeomorphic
2642:
2508:
2504:
2482:
2479:
2476:Definition 3
2463:
2452:
2445:
2138:
2134:
2112:
2109:
1969:
1962:
1664:
1660:
1638:
1635:
1632:Definition 1
1617:
1615:
1612:
1483:
1450:
1428:
1271:
1155:
1005:
1001:
979:
853:
678:
548:
470:
391:
387:
385:
300:
275:
245:and declare
71:
69:
51:
47:
46:is called a
18:
5736:that takes
5159:k-ification
5107:K-ification
4704:composition
4588:restriction
3225:anticompact
2971:Definitions
773:is open in
681:k-open sets
549:k-ification
292:Definitions
176:is open in
6850:Categories
6544:References
6414:Brown 2006
6402:Brown 2006
6197:Brown 2006
6112:Brown 2006
5537:functorial
4501:continuity
4159:sequential
3551:Properties
3487:The space
3440:) and let
2873:Motivation
2503:is called
2133:is called
1659:is called
1562:such that
1370:for every
1250:The space
1000:is called
980:The space
793:for every
734:such that
6718:(1995) .
6498:Rezk 2018
6474:Rezk 2018
6375:Rezk 2018
5987:with the
5811:→
5330:α
5308:α
5281:α
5273:∩
5179:α
4893:⊆
4867:→
4780:→
4728:→
4716:∘
4642:⊆
4616:→
4460:→
4417:×
4381:×
4318:×
4303:with the
4288:…
4235:with the
4220:…
4180:∖
4143:wedge sum
3950:∈
3892:∐
3875:Quotients
3838:−
3790:⊆
3744:→
3668:ω
3637:ω
3607:ω
3579:ω
3563:Subspaces
3555:(See the
3120:⊆
3091:⊆
2695:∅
2622:⊆
2576:∩
2530:⊆
2369:→
2286:−
2238:⊆
2186:→
1987:→
1864:→
1801:⊆
1755:∩
1709:⊆
1626:Hausdorff
1543:⊆
1475:Hausdorff
1381:∈
1335:∩
1289:⊆
810:→
747:−
719:⊆
207:⊆
161:∩
89:⊆
6692:42683260
6666:Topology
6664:(2000).
6637:(1999).
6609:(1998).
6576:(1989).
6553:(2006),
6066:: 1–18.
5998:See also
5101:homotopy
4364:category
4149:Products
3557:Examples
2965:Examples
2887:Hurewicz
2879:k-spaces
2513:coherent
1693:coherent
1276:coherent
1270:is then
359:topology
337:, where
76:coherent
21:topology
6831:at the
6819:at the
6748:0507446
6138:at the
5783:to the
4155:product
3655:is the
3460:be the
3337:product
2883:kompakt
2509:k-space
2457:of the
2139:k-space
1665:k-space
1006:k-space
392:k-space
357:is the
52:k-space
6797:115240
6795:
6785:
6746:
6736:
6690:
6680:
6650:
6623:
6584:
6563:
5916:where
5040:. If
3628:where
2519:a set
2442:space.
2227:a set
1959:space.
1698:a set
6760:(PDF)
6726:Dover
6704:(PDF)
6644:(PDF)
6214:(PDF)
6176:(PDF)
6033:Notes
5947:from
3072:space
2507:or a
2137:or a
2080:into
1928:of a
1924:is a
1663:or a
1004:or a
655:finer
333:be a
6793:OCLC
6783:ISBN
6734:ISBN
6688:OCLC
6678:ISBN
6648:ISBN
6621:ISBN
6582:ISBN
6561:ISBN
5839:The
5394:and
4961:let
4938:and
4153:The
3879:The
3335:The
3320:The
3313:The
3299:The
3212:and
3196:and
3041:For
3021:CG-3
3011:CG-2
3001:CG-1
2984:CG-3
2980:CG-2
2976:CG-1
2867:CGWH
2754:and
2418:(3)
2391:(2)
1935:(4)
1904:(3)
471:The
301:Let
128:open
23:, a
6836:Lab
6824:Lab
6269:doi
6226:doi
6143:Lab
6068:doi
5967:to
5843:in
5779:is
5756:to
5702:to
5135:on
5064:in
5016:to
4800:If
4662:If
4546:If
4503:of
4370:on
3825:to
3464:of
2920:on
2918:Lab
2904:of
1691:is
1477:or
860:in
825:in
685:in
653:is
551:of
503:on
390:or
130:in
126:is
63:or
50:or
19:In
6852::
6791:.
6781:.
6777::
6773:.
6744:MR
6742:.
6732:.
6714:;
6686:.
6676:.
6672::
6382:^
6265:23
6263:.
6259:.
6222:88
6220:.
6216:.
6189:^
6149:^
6104:^
6062:.
5991:.
4141:A
3997:A
3394:).
3365:).
3310:).
3063:.
3059:,
3055:,
2982:,
2978:,
2961:.
2924:.
2472:.
1628:.
288:.
273:.
6834:n
6822:n
6799:.
6762:.
6750:.
6724:(
6706:.
6694:.
6656:.
6629:.
6617:5
6590:.
6340:.
6277:.
6271::
6232:.
6228::
6184:)
6180:(
6178:.
6141:n
6126:.
6074:.
6070::
6064:9
5975:Y
5955:X
5929:X
5925:Y
5904:)
5899:X
5895:Y
5891:(
5888:k
5867:s
5864:u
5861:a
5858:H
5855:G
5852:C
5825:.
5821:p
5818:o
5815:T
5807:p
5804:o
5801:T
5798:G
5795:C
5767:X
5764:k
5744:X
5723:p
5720:o
5717:T
5714:G
5711:C
5689:p
5686:o
5683:T
5661:p
5658:o
5655:T
5652:G
5649:C
5627:s
5624:u
5621:a
5618:H
5615:G
5612:C
5589:p
5586:o
5583:T
5560:p
5557:o
5554:T
5551:G
5548:C
5520:X
5500:X
5497:k
5477:.
5474:X
5471:=
5468:X
5465:k
5445:X
5425:X
5422:k
5402:X
5382:X
5379:k
5359:.
5356:X
5353:k
5333:.
5304:K
5277:K
5270:A
5247:A
5227:X
5207:.
5204:X
5184:}
5175:K
5171:{
5143:X
5119:X
5087:)
5084:Y
5081:,
5078:X
5075:(
5072:C
5048:X
5024:Y
5004:X
4984:)
4981:Y
4978:,
4975:X
4972:(
4969:C
4949:,
4946:Y
4926:X
4899:.
4896:X
4890:K
4870:Y
4864:K
4861::
4856:K
4852:|
4848:f
4828:f
4808:X
4786:.
4783:X
4777:K
4774::
4771:u
4751:K
4731:Y
4725:K
4722::
4719:u
4713:f
4690:f
4670:X
4648:.
4645:X
4639:K
4619:Y
4613:K
4610::
4605:K
4601:|
4597:f
4574:f
4554:X
4531:f
4511:f
4483:X
4463:Y
4457:X
4454::
4451:f
4423:)
4420:Y
4414:X
4411:(
4408:k
4384:Y
4378:X
4321:Y
4315:X
4291:}
4285:,
4282:3
4279:,
4276:2
4273:,
4270:1
4267:{
4263:/
4258:R
4254:=
4251:Y
4223:}
4217:,
4214:3
4210:/
4206:1
4203:,
4200:2
4196:/
4192:1
4189:,
4186:1
4183:{
4176:R
4172:=
4169:X
4118:]
4115:1
4112:,
4109:0
4106:(
4086:]
4083:1
4080:,
4077:0
4074:[
4040:.
4037:X
4013:X
3980:i
3976:X
3953:I
3947:i
3943:)
3937:i
3933:X
3929:(
3907:i
3903:X
3897:i
3852:)
3849:U
3846:(
3841:1
3834:q
3813:q
3793:X
3787:U
3767:Y
3747:X
3741:Y
3738::
3735:q
3715:,
3712:X
3672:1
3641:1
3616:]
3611:1
3603:,
3600:0
3597:[
3594:=
3591:1
3588:+
3583:1
3535:X
3515:Y
3495:X
3475:.
3472:Y
3448:X
3427:R
3406:Y
3377:Z
3348:R
3324:.
3317:.
3293:1
3279:X
3259:X
3249:1
3235:X
3180:1
3166:X
3146:;
3143:K
3123:X
3117:K
3097:,
3094:X
3088:}
3085:x
3082:{
3070:1
3068:T
2916:n
2843:.
2840:]
2837:1
2834:,
2831:0
2828:(
2808:]
2805:1
2802:,
2799:0
2796:[
2768:}
2765:1
2762:{
2742:}
2739:0
2736:{
2716:}
2713:X
2710:,
2707:}
2704:1
2701:{
2698:,
2692:{
2672:}
2669:1
2666:,
2663:0
2660:{
2657:=
2654:X
2628:.
2625:X
2619:K
2599:K
2579:K
2573:A
2553:X
2533:X
2527:A
2491:X
2426:X
2399:X
2375:.
2372:X
2366:K
2363::
2360:f
2340:K
2320:K
2300:)
2297:A
2294:(
2289:1
2282:f
2261:X
2241:X
2235:A
2212:.
2209:K
2189:X
2183:K
2180::
2177:f
2153:X
2121:X
2091:.
2088:X
2068:)
2065:K
2062:(
2059:f
2039:)
2036:K
2033:(
2030:f
2010:K
1990:X
1984:K
1981::
1978:f
1943:X
1912:X
1890:.
1887:K
1867:X
1861:K
1858::
1855:f
1831:X
1807:.
1804:X
1798:K
1778:K
1758:K
1752:A
1732:X
1712:X
1706:A
1679:X
1647:X
1597:.
1592:G
1570:X
1548:F
1538:G
1514:F
1492:X
1461:X
1437:X
1414:,
1411:X
1391:.
1386:C
1378:K
1358:K
1338:K
1332:A
1312:X
1292:X
1286:A
1258:X
1238:.
1235:X
1213:C
1191:,
1188:X
1166:F
1142:.
1139:X
1136:=
1133:X
1130:k
1110:;
1107:X
1087:,
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1064:X
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1018:F
988:X
966:.
963:X
960:k
937:F
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840:.
835:F
813:X
807:K
804::
801:f
781:K
761:)
758:U
755:(
750:1
743:f
722:X
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696:;
693:X
665:T
641:T
621:,
618:)
615:T
612:,
609:X
606:(
584:F
562:.
559:T
533:F
511:X
488:F
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456:,
451:F
429:.
426:X
404:F
372:.
369:X
345:T
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318:T
315:,
312:X
309:(
253:X
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213:.
210:X
204:K
184:K
164:K
158:A
138:X
114:A
95:,
92:X
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