1113:
73:
direction or + direction) is still preserved on the circle; for if a number approaches towards infinity from the - direction on the number line, then the corresponding point on the circle can approach ā from the left for example. Then if a number approaches infinity from the + direction on the number line, then the corresponding point on the circle can approach ā from the right.
72:
to a circle in the plane (which, as a closed and bounded subset of the
Euclidean plane, is compact). Every sequence that ran off to infinity in the real line will then converge to ā in this compactification. The direction in which a number approaches infinity on the number line (either in the -
210:, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
67:
with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ā. The resulting compactification is
813:
54:
of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
636:ā the cohomology of affine space is trivial, while the cohomology of projective space is non-trivial and reflects the key features of intersection theory (dimension and degree of a subvariety, with intersection being
820:
101:-direction) and move them towards each other, until you get a circle with one point (the topmost one) missing. This point is our new point ā "at infinity"; adding it in completes the compact circle.
914:, their additional linear structure allowing e.g. for developing a differential calculus and more advanced considerations e.g. in relaxation in variational calculus or optimization theory.
651:
generally require allowing certain degeneracies ā for example, allowing certain singularities or reducible varieties. This is notably used in the
DeligneāMumford compactification of the
816:
848:
213:
The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.
726:
295:
824:
1116:
1075:
557:
can "escape", one new point at infinity is added (but each direction is identified with its opposite). The
Alexandroff one-point compactification of
1129:
628:, holds in projective space but not affine space. This distinct behavior of intersections in affine space and projective space is reflected in
706:, and those lattices can degenerate ('go off to infinity'), often in a number of ways (taking into account some auxiliary structure of
17:
1167:
612:
because the added points at infinity lead to simpler formulations of many theorems. For example, any two different lines in
339:
318:
1145:
961:
893:
155:
of the real line, discussed in more generality below. It is also possible to compactify the real line by adding
370:
323:
Of particular interest are
Hausdorff compactifications, i.e., compactifications in which the compact space is
532:
882:
904:
652:
528:
222:
808:{\displaystyle {\text{SO}}(n)\setminus {\text{SL}}_{n}({\textbf {R}})/{\text{SL}}_{n}({\textbf {Z}}).}
897:
672:
875:
583:
512:
910:
The compactifications that are simultaneously convex subsets in a locally convex space are called
274:
911:
527:
in real projective four-space. The method is similar to that used to provide a base manifold for
520:
489:
355:
351:
621:
1025:
986:
924:
867:
543:
151:
between the real line and the unit circle without ā. What we have constructed is called the
860:
1007:
496:
as a subset of that space will also be compact. This is the StoneāÄech compactification.
147:
Since tangents and inverse tangents are both continuous, our identification function is a
8:
625:
1094:
1052:
691:
629:
609:
366:
160:
76:
Intuitively, the process can be pictured as follows: first shrink the real line to the
1141:
1070:
977:
957:
871:
398:
192:
43:
815:
This is harder to compactify. There are a variety of compactifications, such as the
1084:
1042:
1034:
1003:
995:
886:
566:
35:
637:
856:
852:
720:
703:
702:). Here the cusps are there for a good reason: the curves parametrize a space of
699:
695:
633:
328:
324:
306:
207:
203:
837:
664:
602:
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1020:
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The StoneāÄech compactification can be constructed explicitly as follows: let
1161:
1137:
1112:
953:
945:
687:
335:
196:
148:
77:
69:
47:
327:. A topological space has a Hausdorff compactification if and only if it is
1073:(1937), "Applications of the theory of Boolean rings to general topology",
648:
508:
641:
516:
504:
105:
31:
658:
206:
may be of particular interest. Since every compact
Hausdorff space is a
1098:
1056:
999:
51:
841:
668:
180:
172:
144:/2) is undefined; we will identify this point with our point ā.
64:
1089:
1038:
710:). The cusps stand in for those different 'directions to infinity'.
27:
Embedding a topological space into a compact space as a dense subset
982:"Ćber die Metrisation der im Kleinen kompakten topologischen RƤume"
252:
616:
intersect in precisely one point, a statement that is not true in
524:
129:
97:-axis; then bend the ends of this interval upwards (in positive
981:
683:
to preserve structure at a richer level than just topological.
561:
we constructed in the example above is in fact homeomorphic to
431:
is a compact
Hausdorff space, there is a unique continuous map
113:
676:
361:"Most general" or formally "reflective" means that the space
332:
109:
128:
on the circle with the corresponding point on the real line
690:
are compactified by the addition of single points for each
1134:
593:; the Alexandroff one-point compactification of the plane
358:
of the category of
Tychonoff spaces and continuous maps.
297:
is closed and compact. The one-point compactification of
216:
199:, because of the special properties compact spaces have.
247:
is obtained by adding one extra point ā (often called a
1110:
15 parameter conformal group of spacetime described in
82:
927: ā Way to extend a non-compact topological space
729:
659:
Compactification and discrete subgroups of Lie groups
601:, which in turn can be identified with a sphere, the
354:
of
Compact Hausdorff spaces and continuous maps as a
277:
723:the same questions can be posed, for example about
553:. For each possible "direction" in which points in
830:
807:
480:can be identified with a subset of , the space of
289:
1076:Transactions of the American Mathematical Society
597:is (homeomorphic to) the complex projective line
472:can be identified with an evaluation function on
338:) "most general" Hausdorff compactification, the
104:A bit more formally: we represent a point on the
1159:
608:Passing to projective space is a common tool in
381:can be extended to a continuous function from
312:
136:/2). This function is undefined at the point
499:
468:to the closed interval . Then each point in
1117:Associative Composition Algebra/Homographies
576:the one-point compactification of the plane
50:. A compact space is a space in which every
976:
713:That is all for lattices in the plane. In
521:compactification for split complex numbers
124:for simplicity. Identify each such point
1088:
1046:
549:is a compactification of Euclidean space
1128:
523:. In fact, the hyperboloid is part of a
464:be the set of continuous functions from
393:is a compact Hausdorff space containing
255:of the new space to be the open sets of
944:
159:points, +ā and āā; this results in the
14:
1160:
821:reductive BorelāSerre compactification
217:Alexandroff one-point compactification
183:subset of a compact space is called a
153:Alexandroff one-point compactification
1069:
847:Some 'boundary' theories such as the
679:is often a candidate for more subtle
488:to . Since the latter is compact by
409:is the same as the given topology on
227:For any noncompact topological space
42:is the process or result of making a
1019:
580:since more than one point is added.
794:
764:
538:
331:. In this case, there is a unique (
259:together with the sets of the form
24:
907:of a quotient of algebraic groups.
389:in a unique way. More explicitly,
25:
1179:
874:arises from the consideration of
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698:(and so, since they are compact,
281:
1111:
653:moduli space of algebraic curves
1023:(1937). "On bicompact spaces".
831:Other compactification theories
589:is also a compactification of
1168:Compactification (mathematics)
1122:
1104:
1063:
1013:
970:
938:
799:
789:
769:
759:
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350:; formally, this exhibits the
191:. It is often useful to embed
13:
1:
931:
849:collaring of an open manifold
413:, and for any continuous map
377:to a compact Hausdorff space
166:
58:
894:BailyāBorel compactification
817:BorelāSerre compactification
533:conformal group of spacetime
301:is Hausdorff if and only if
290:{\displaystyle X\setminus G}
7:
918:
883:projective line over a ring
340:StoneāÄech compactification
319:StoneāÄech compactification
313:StoneāÄech compactification
10:
1184:
905:wonderful compactification
624:, which is fundamental in
500:Spacetime compactification
316:
237:one-point compactification
223:One-point compactification
220:
18:Conformal compactification
898:Hermitian symmetric space
876:almost periodic functions
519:can be used to provide a
912:convex compactifications
825:Satake compactifications
584:Complex projective space
565:. Note however that the
513:stereographic projection
365:is characterized by the
263: āŖ {ā}, where
202:Embeddings into compact
175:of a topological space
827:, that can be formed.
809:
356:reflective subcategory
291:
1026:Annals of Mathematics
987:Mathematische Annalen
978:Alexandroff, Pavel S.
925:Alexandroff extension
868:Bohr compactification
810:
544:Real projective space
292:
267:is an open subset of
861:Furstenberg boundary
727:
647:Compactification of
515:onto a single-sheet
453:is identically
275:
896:of a quotient of a
626:intersection theory
490:Tychonoff's theorem
371:continuous function
251:) and defining the
1071:Stone, Marshall H.
1048:10338.dmlcz/100420
1000:10.1007/BF01448011
889:may compactify it.
805:
630:algebraic topology
620:. More generally,
610:algebraic geometry
367:universal property
287:
193:topological spaces
161:extended real line
946:Munkres, James R.
872:topological group
796:
781:
766:
751:
733:
492:, the closure of
305:is Hausdorff and
249:point at infinity
44:topological space
16:(Redirected from
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994:(3ā4): 294ā301,
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952:(2nd ed.).
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887:topological ring
836:The theories of
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700:algebraic curves
696:Riemann surfaces
681:compactification
663:In the study of
634:cohomology rings
622:BĆ©zout's theorem
567:projective plane
539:Projective space
444:
426:
399:induced topology
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294:
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204:Hausdorff spaces
185:compactification
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853:Martin boundary
838:ends of a space
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721:Euclidean space
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511:have shown how
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484:functions from
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307:locally compact
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688:modular curves
673:quotient space
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603:Riemann sphere
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449:restricted to
397:such that the
317:Main article:
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221:Main article:
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197:compact spaces
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116:, going from ā
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686:For example,
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667:subgroups of
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649:moduli spaces
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63:Consider the
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1130:RoubĆÄek, T.
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1119:at Wikibooks
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1021:Äech, Eduard
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140:, since tan(
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642:cup product
517:hyperboloid
505:Walter Benz
233:Alexandroff
106:unit circle
32:mathematics
1136:. Berlin:
1008:50.0128.04
932:References
842:prime ends
823:, and the
669:Lie groups
445:for which
271:such that
167:Definition
59:An example
52:open cover
745:∖
369:that any
329:Tychonoff
325:Hausdorff
282:∖
253:open sets
173:embedding
65:real line
1162:Category
1132:(1997).
980:(1924),
950:Topology
948:(2000).
919:See also
704:lattices
665:discrete
476:. Thus
436: :
427:, where
418: :
352:category
1099:1989788
1057:1968839
640:to the
632:in the
531:of the
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108:by its
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333:up to
231:the (
181:dense
179:as a
112:, in
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881:The
866:The
859:and
840:and
692:cusp
507:and
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482:all
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120:to
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