387:. An alternative approach, equivalent to the one given here, is first to define "Polish metric space" to mean "complete separable metric space", and then to define a "Polish space" as the topological space obtained from a Polish metric space by
320:. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology.
363:
452:) if it is the image of a Polish space under a continuous mapping. So every Lusin space is Suslin. In a Polish space, a subset is a Suslin space if and only if it is a
639:
is an open mapping. As a result, it is a remarkable fact about Polish groups that Baire-measurable mappings (i.e., for which the preimage of any open set has the
381:
giving rise to the same topology, but no one of these is singled out or distinguished. A Polish space with a distinguished complete metric is called a
338:
A second characterization follows from
Alexandrov's theorem. It states that a separable metric space is completely metrizable if and only if it is a
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900:
620:
1233:
573:
that is also a Polish space, in other words homeomorphic to a separable complete metric space. There are several classic results of
1098:
1069:
1204:
1174:
1130:
987:
55:
subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—
317:
1223:
316:
There are numerous characterizations that tell when a second-countable topological space is metrizable, such as
973:
1196:
1162:
1086:
105:. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the
647:
is a universal Polish group, in the sense that every Polish group is isomorphic to a closed subgroup of it.
208:
128:
31:
643:) that are homomorphisms between them are automatically continuous. The group of homeomorphisms of the
72:
67:
and others. However, Polish spaces are mostly studied today because they are the primary setting for
1228:
666:
102:
237:
is uncountable, it can be written as the disjoint union of a perfect set and a countable open set.
1080:
992:
473:
countable intersections or countable unions of Suslin subspaces of a
Hausdorff topological space,
341:
526:
68:
522:
1184:
961:
688:
624:
582:
331:. A separable metric space is completely metrizable if and only if the second player has a
127:
that preserves the Borel structure. In particular, every uncountable Polish space has the
8:
512:
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949:
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80:
1005:
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Diffusions, Markov
Processes, and Martingales, Volume 1: Foundations, 2nd Edition
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916:
534:
426:
378:
286:
91:
28:
714:
449:
324:
224:
76:
42:
678:
The group of isometries of a separable complete metric space is a Polish group
422:
A subspace of a Lusin space is a Lusin space if and only if it is a Borel set.
323:
There is a characterization of complete separable metric spaces in terms of a
1217:
1020:
933:
882:
662:
574:
530:
509:
409:
296:
countable intersections of Polish subspaces of a
Hausdorff topological space,
106:
64:
627:
due to
Kuratowski: a continuous injective homomorphism of a Polish subgroup
56:
644:
585:
on homomorphisms between Polish groups. Firstly, Banach's argument applies
538:
501:
374:
328:
248:
98:
94:
45:
38:
435:
The disjoint union of a countable number of Lusin spaces is a Lusin space.
373:
Although Polish spaces are metrizable, they are not in and of themselves
228:
116:
1123:
Gradient Flows in Metric Spaces and in the Space of
Probability Measures
1120:
672:
The group of homeomorphisms of a compact metric space is a Polish group.
1062:
Radon
Measures on Arbitrary Topological Spaces and Cylindrical Measures
1040:
1015:
953:
928:
453:
60:
675:
The product of a countable number of Polish groups is a Polish group.
655:
124:
87:
52:
49:
1032:
945:
283:
products and disjoint unions of countable families of Polish spaces,
1016:"On continuity and openness of homomorphisms in topological groups"
432:
The product of a countable number of Lusin spaces is a Lusin space.
988:"Group extensions and cohomology for locally compact groups. III"
75:. Polish spaces are also a convenient setting for more advanced
525:. Since a probability measure is globally finite, and hence a
425:
Any countable union or intersection of Lusin subspaces of a
412:) if some stronger topology makes it into a Polish space.
853:
778:, pp. 94, 102, Lemma 4 and Corollary 1 of Theorem 5.
415:
There are many ways to form Lusin spaces. In particular:
658:
with a countable number of components are Polish groups.
470:
countable products and disjoint unions of Suslin spaces,
911:(1989). "IX. Use of Real Numbers in General Topology".
529:, every probability measure on a Radon space is also a
179:
is Polish (under the induced topology) if and only if
344:
185:
is the intersection of a sequence of open subsets of
829:
619:is continuous. Secondly, there is a version of the
394:
1188:
865:
357:
231:and a countable set. Further, if the Polish space
1121:Ambrosio, L., Gigli, N. & Savaré, G. (2005).
913:Elements of Mathematics: General Topology, Part 2
365:subset of its completion in the original metric.
1215:
1047:
889:. Monografie Matematyczne (in French). Warsaw.
164:(by virtue of being separable and metrizable).
1145:: CS1 maint: multiple names: authors list (
1183:
923:
847:
1078:
1048:Rogers, L. C. G.; Williams, David (1994).
960:
899:: CS1 maint: location missing publisher (
859:
739:
1004:
929:"Einige Sätze ueber topologische Gruppen"
706:
607:Polish, then any Borel homomorphism from
467:closed or open subsets of a Suslin space,
86:Common examples of Polish spaces are the
1125:. Basel: ETH Zürich, Birkhäuser Verlag.
1056:
907:
811:
799:
787:
775:
766:, p. 102, Corollary 1 to Theorem 5.
763:
751:
727:
240:Every Polish space is homeomorphic to a
1153:
368:
307:the standard topology of the real line.
1216:
1013:
881:
871:
835:
151:are generalizations of Polish spaces.
982:
823:
555:Every Suslin space is a Radon space.
483:They have the following properties:
217:is Polish then any closed subset of
802:, p. 95, Corollary of Lemma 5.
476:continuous images of Suslin spaces,
444:A Hausdorff topological space is a
419:Every Polish space is a Lusin space
404:A Hausdorff topological space is a
311:
13:
1114:
141:
19:In the mathematical discipline of
14:
1245:
1006:10.1090/S0002-9947-1976-0414775-X
661:The unitary group of a separable
601:are separable metric spaces with
589:to non-Abelian Polish groups: if
463:The following are Suslin spaces:
277:closed subsets of a Polish space,
273:The following spaces are Polish:
135:
1234:Science and technology in Poland
1191:Classical Descriptive Set Theory
1079:Srivastava, Sashi Mohan (1998).
887:Théorie des opérations linéaires
558:
487:Every Suslin space is separable.
479:Borel subsets of a Suslin space.
439:
395:Generalizations of Polish spaces
377:; each Polish space admits many
841:
817:
491:
399:
289:spaces that are metrizable and
280:open subsets of a Polish space,
269:is the set of natural numbers).
805:
793:
781:
769:
757:
745:
733:
721:
707:Gemignani, Michael C. (1967).
700:
1:
1197:Graduate Texts in Mathematics
1163:Graduate Texts in Mathematics
1087:Graduate Texts in Mathematics
694:
318:Urysohn's metrization theorem
154:
1159:An Invitation to C*-Algebras
1052:. John Wiley & Sons Ltd.
129:cardinality of the continuum
7:
1199:. Vol. 156. Springer.
1064:. Oxford University Press.
682:
358:{\displaystyle G_{\delta }}
73:Borel equivalence relations
10:
1252:
1165:. Vol. 39. New York:
826:, p. 8, Proposition 5
633:onto another Polish group
119:Polish spaces, there is a
713:. Internet Archive. USA:
263:is the unit interval and
71:, including the study of
667:strong operator topology
209:Cantor–Bendixson theorem
993:Trans. Amer. Math. Soc.
790:, pp. 95, Lemma 6.
654:All finite dimensional
567:is a topological group
1224:Descriptive set theory
1082:A Course on Borel Sets
1014:Pettis, B. J. (1950).
527:locally finite measure
359:
223:can be written as the
160:Every Polish space is
69:descriptive set theory
360:
291:countable at infinity
32:completely metrizable
689:Standard Borel space
669:) is a Polish group.
625:closed graph theorem
621:open mapping theorem
369:Polish metric spaces
342:
327:known as the strong
717:. pp. 142–143.
710:Elementary Topology
513:probability measure
384:Polish metric space
79:, in particular in
37:; that is, a space
16:Concept in topology
968:. Academic Press.
814:, pp. 197–199
552:is a Radon space.
533:. In particular a
355:
303:with the topology
301:irrational numbers
173:of a Polish space
81:probability theory
1100:978-0-387-98412-4
1058:Schwartz, Laurent
925:Freudenthal, Hans
909:Bourbaki, Nicolas
641:property of Baire
506:topological space
456:(an image of the
429:is a Lusin space.
121:Borel isomorphism
35:topological space
1241:
1229:General topology
1210:
1194:
1180:
1155:Arveson, William
1150:
1144:
1136:
1110:
1108:
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984:Moore, Calvin C.
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898:
890:
875:
869:
863:
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851:
848:Freudenthal 1936
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587:mutatis mutandis
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458:Suslin operation
379:complete metrics
364:
362:
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333:winning strategy
312:Characterization
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162:second countable
115:Between any two
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21:general topology
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1177:
1167:Springer-Verlag
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1115:Further reading
1105:
1103:
1101:
1091:Springer-Verlag
1072:
1033:10.2307/1969471
976:
966:Topology Vol. I
946:10.2307/1968686
917:Springer-Verlag
892:
891:
878:
870:
866:
860:Kuratowski 1966
858:
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842:
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818:
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740:Srivastava 1998
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697:
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541:
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508:on which every
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442:
427:Hausdorff space
402:
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349:
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340:
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287:locally compact
264:
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247:-subset of the
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202:
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17:
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1116:
1113:
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1111:
1099:
1076:
1071:978-0195605167
1070:
1054:
1045:
1027:(2): 293–308.
1011:
980:
974:
962:Kuratowski, K.
958:
921:
905:
883:Banach, Stefan
877:
876:
864:
862:, p. 400.
852:
840:
828:
816:
804:
792:
780:
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744:
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720:
715:Addison-Wesley
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500:, named after
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450:Mikhail Suslin
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335:in this game.
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281:
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271:
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225:disjoint union
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77:measure theory
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1021:Ann. of Math.
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998:
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934:Ann. of Math.
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910:
906:
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856:
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838:, p. 23.
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832:
825:
820:
813:
812:Bourbaki 1989
808:
801:
800:Schwartz 1973
796:
789:
788:Schwartz 1973
784:
777:
776:Schwartz 1973
772:
765:
764:Schwartz 1973
760:
753:
752:Schwartz 1973
748:
741:
736:
730:, p. 197
729:
728:Bourbaki 1989
724:
716:
712:
711:
703:
699:
690:
687:
686:
677:
674:
671:
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664:
663:Hilbert space
660:
657:
653:
652:
651:
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646:
642:
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631:
626:
622:
617:
611:
605:
599:
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588:
584:
580:
576:
571:
566:
559:Polish groups
556:
553:
549:
545:
540:
536:
532:
531:Radon measure
528:
524:
523:inner regular
519:
514:
511:
507:
503:
499:
486:
485:
484:
478:
475:
472:
469:
466:
465:
464:
461:
459:
455:
451:
448:(named after
447:
440:Suslin spaces
434:
431:
428:
424:
421:
418:
417:
416:
413:
411:
410:Nikolai Lusin
408:(named after
407:
392:
390:
386:
385:
380:
376:
375:metric spaces
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350:
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302:
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251:(that is, of
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144:
143:
142:Suslin spaces
138:
137:
132:
130:
126:
123:; that is, a
122:
118:
113:
108:
107:open interval
104:
100:
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89:
84:
82:
78:
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70:
66:
62:
58:
54:
51:
47:
44:
40:
36:
33:
30:
26:
22:
1190:
1158:
1122:
1104:. Retrieved
1081:
1061:
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1024:
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996:
991:
965:
940:(1): 46–56.
937:
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912:
886:
867:
855:
850:, p. 54
843:
831:
819:
807:
795:
783:
771:
759:
754:, p. 94
747:
742:, p. 55
735:
723:
709:
702:
649:
645:Hilbert cube
635:
629:
615:
609:
603:
597:
591:
586:
569:
565:Polish group
564:
562:
554:
547:
543:
539:metric space
517:
502:Johann Radon
497:
495:
492:Radon spaces
482:
462:
446:Suslin space
445:
443:
414:
405:
403:
400:Lusin spaces
391:the metric.
383:
382:
372:
337:
329:Choquet game
322:
315:
272:
265:
259:
253:
249:Hilbert cube
233:
219:
213:
193:
187:
181:
175:
169:
148:Radon spaces
146:
140:
136:Lusin spaces
134:
133:
114:
99:Cantor space
95:Banach space
85:
46:metric space
39:homeomorphic
25:Polish space
24:
18:
1185:Kechris, A.
872:Pettis 1950
836:Banach 1932
579:Freudenthal
498:Radon space
406:Lusin space
299:the set of
229:perfect set
167:A subspace
117:uncountable
112:is Polish.
103:Baire space
48:that has a
1218:Categories
1106:2008-12-04
975:012429202X
824:Moore 1976
695:References
665:(with the
656:Lie groups
650:Examples:
583:Kuratowski
454:Suslin set
389:forgetting
305:induced by
155:Properties
101:, and the
61:Kuratowski
57:Sierpiński
1141:cite book
895:cite book
537:complete
535:separable
351:δ
125:bijection
92:separable
88:real line
50:countable
29:separable
1187:(1995).
1157:(1981).
1060:(1973).
999:: 1–33.
986:(1976).
964:(1966).
927:(1936).
885:(1932).
683:See also
257:, where
43:complete
1041:1969471
954:1968686
623:or the
504:, is a
1203:
1173:
1129:
1097:
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1039:
972:
952:
575:Banach
204:-set).
145:, and
110:(0, 1)
97:, the
90:, any
65:Tarski
1037:JSTOR
950:JSTOR
510:Borel
227:of a
211:) If
197:is a
53:dense
41:to a
27:is a
1201:ISBN
1171:ISBN
1147:link
1127:ISBN
1095:ISBN
1066:ISBN
970:ISBN
901:link
595:and
581:and
325:game
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1029:doi
1001:doi
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613:to
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515:on
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