4710:
and paracompactness: For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite",
5129:. The proof that all metrizable spaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces full normality and paracompactness are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later
4156:
3378:(by Urysohn's lemma for disjoint closed sets in normal spaces, which a paracompact Hausdorff space is). Note by the support of a function, we here mean the points not mapping to zero (and not the closure of this set). To show that
2930:
1307:, that is, every open cover of a paracompact Hausdorff space has a shrinking: another open cover indexed by the same set such that the closure of every set in the new cover lies inside the corresponding set in the old cover.
909:
2110:
4939:
3932:
73:
of a paracompact space is paracompact. While compact subsets of
Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called
3870:
376:
5093:
4660:
4251:
3376:
1789:
3333:
2616:
1843:
3430:
2732:
2242:
1661:
679:
765:
539:
257:
488:
2348:
1611:
4579:
4617:
4320:
2972:
3182:
3040:
2416:
4282:
1568:
4539:
4079:
5276:
2791:
2165:
3570:
3524:
2300:
2267:
2135:
2031:
2006:
1957:
1932:
1897:
1694:
1509:
4691:
4035:
3213:
2832:
2513:
2440:
1981:
4208:
3277:
1746:
5474:
3732:
3668:
3613:
619:
569:
4417:
2998:
2374:
5369:
5334:
5249:
4465:
3786:
3457:
4381:
1537:
3983:
3762:
4353:
3813:
2670:
2643:
2192:
4960:
is a cover of the same space such that, given any point in the space, the star of the point in the new cover is a subset of some set in the old cover. In symbols,
4507:
4486:
4438:
4004:
3953:
3689:
3634:
3545:
3499:
3478:
1872:
1469:
4084:
3140:
3120:
3100:
3080:
3060:
2761:
2542:
2489:
2469:
2033:
by a locally finite open refinement. One can easily check that each set in this refinement has the same property as that which characterised the original cover.
1039:
1019:
999:
979:
959:
932:
845:
825:
805:
785:
702:
639:
589:
433:
399:
321:
301:
281:
206:
182:
158:
2837:
4714:
Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.
1217:
is paracompact if every open cover admits a locally finite refinement. (Here, the refinement is not required to be open.) In particular, every regular
5675:
853:
2041:
105:
of any collection of compact topological spaces is compact. However, the product of a paracompact space and a compact space is always paracompact.
4867:
5122:
Without the
Hausdorff property, paracompact spaces are not necessarily fully normal. Any compact space that is not regular provides an example.
3875:
5581:
5514:
1425:
Partitions of unity are useful because they often allow one to extend local constructions to the whole space. For instance, the integral of
5241:
5670:
4785:
if every open cover has an open refinement such that the intersection of all the open sets about any point in this refinement is open.
5612:
5554:
5424:
3818:
329:
5195:
5043:
4765:
There are several variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:
1983:. One can check as an exercise that this provides an open refinement, since paracompact Hausdorff spaces are regular, and since
1205:
Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This can be extended to
1115:
4821:" can be added to any of the adjectives "paracompact", "metacompact", and "fully normal" to make the requirement apply only to
4622:
4213:
3338:
1751:
3282:
1340:
1041:. Note that an open cover on a topological space is locally finite iff its a locally finite cover of the underlying locale.
2547:
1794:
5648:
3381:
5115:
space is paracompact. In fact, for
Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T
2675:
5643:
5540:
4745:
A paracompact subset of a
Hausdorff space need not be closed. In fact, for metric spaces, all subsets are paracompact.
2197:
1616:
644:
4841:
of the point in the cover is the union of all the sets in the cover that contain the point. In symbols, the star of
719:
493:
211:
5665:
1437:
and the integral is well known), and this definition is then extended to the whole space via a partition of unity.
1418:
space is
Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover (see
592:
442:
2310:
1573:
1231:
1157:
1119:
40:
4544:
4584:
4287:
2935:
542:
117:
3145:
3003:
2379:
709:
97:
is a paracompact locale, but the product of two paracompact spaces may not be paracompact. Compare this to
5447:. Annals of Mathematics Studies. Vol. 2. Princeton University Press, Princeton, N. J. pp. ix+90.
4256:
1542:
5569:
5535:
4512:
4052:
1150:
1071:
98:
2766:
2140:
1372:
1228:) A topological space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable.
5638:
5312:
63:
subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be
Hausdorff.
3550:
3504:
2272:
2247:
2115:
2011:
1986:
1937:
1912:
1877:
1674:
1489:
4675:
4019:
3197:
2796:
1135:
1131:
1081:
is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable.
94:
2494:
2421:
1962:
5146:
4161:
3230:
1699:
70:
4945:
The notation for the star is not standardised in the literature, and this is just one possibility.
3694:
3639:
3575:
1422:). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).
5573:
4781:
1332:
941:
This definition extends verbatim to locales, with the exception of locally finite: an open cover
597:
185:
133:
102:
90:
86:
60:
1315:
1168:
547:
4386:
2977:
2353:
1471:
is paracompact if and only if it every open cover admits a subordinate partition of unity. The
5608:
5587:
5577:
5550:
5531:
5520:
5510:
5483:
5469:
5420:
5268:
4810:
4773:
4443:
3767:
3435:
1426:
1292:
1258:
1218:
1186:
1061:
138:
28:
5414:
4358:
1514:
5378:
5343:
5325:
5258:
5210:
5151:
5134:
4749:
3958:
3737:
1339:
is a paracompact
Hausdorff space with a given open cover, then there exists a collection of
1311:
1193:
1142:
1103:
113:
36:
5623:
5495:
5452:
5125:
A historical note: fully normal spaces were defined before paracompact spaces, in 1940, by
5028:) if every point of the space belongs to only finitely many sets in the cover. In symbols,
4329:
4151:{\displaystyle \{V{\text{ open }}:(\exists {U\in {\mathcal {O}}}){\bar {V}}\subseteq U\}\,}
3791:
2648:
2621:
2170:
5546:
5491:
5448:
4952:
4792:
4730:
1434:
1328:
1304:
1282:
1246:
Although a product of paracompact spaces need not be paracompact, the following are true:
1182:
1146:
1111:
1107:
1078:
1068:
1065:
120:
52:
4491:
4470:
4422:
3988:
3937:
3673:
3618:
3529:
3483:
3462:
1856:
1453:
1102:
is paracompact. Early proofs were somewhat involved, but an elementary one was found by
16:
Topological space in which every open cover has an open refinement that is locally finite
1402:
such that all but finitely many of the functions in the collection are identically 0 in
5502:
5130:
3125:
3105:
3085:
3065:
3045:
2925:{\displaystyle {\bar {W_{U}}}\subseteq {\bar {A_{1}}}\cup ...\cup {\bar {A_{n}}}\cup C}
2737:
2518:
2474:
2445:
1153:
1024:
1004:
984:
964:
944:
917:
830:
810:
790:
770:
687:
624:
574:
418:
384:
306:
286:
266:
260:
191:
167:
143:
5601:
5348:
5329:
5263:
1419:
5659:
5561:
5292:
5126:
4828:
Every paracompact space is metacompact, and every metacompact space is orthocompact.
4707:
1348:
1251:
1214:
1058:
1051:
48:
5214:
1238:
into nonempty closed convex subsets of Banach spaces admit continuous selection iff
5100:
4734:
1297:
1099:
109:
56:
5383:
5364:
1206:
1161:
1094:
20:
1174:
is a non-paracompact surface. (It is easy to find an uncountable open cover of
5440:
1266:
1085:
904:{\displaystyle \left\{\alpha \in A:U_{\alpha }\cap V\neq \varnothing \right\}}
713:
403:
67:
32:
5524:
5487:
5272:
4419:, we have now — all things remaining the same — that their sum is everywhere
2105:{\displaystyle W_{U}=\bigcup \{A\in {\mathcal {V}}:{\bar {A}}\subseteq U\}\,}
5591:
4934:{\displaystyle \mathbf {U} ^{*}(x):=\bigcup _{U_{\alpha }\ni x}U_{\alpha }.}
4822:
4662:. Thus we have a partition of unity subordinate to the original open cover.
1899:
is an open cover, then there exists a partition of unity subordinate to it.
435:
is a new cover of the same space such that every set in the new cover is a
4803:
3927:{\displaystyle f\upharpoonright N=\sum _{U\in S}f_{U}\upharpoonright N\,}
1430:
408:
79:
5300:
5396:
5394:
4722:
Paracompactness is similar to compactness in the following respects:
1196:
is not paracompact despite being a product of two paracompact spaces.
436:
161:
1696:
is a locally finite open cover, then there are continuous functions
4253:(thus the usual closed version of the support is contained in some
1934:
be the collection of open sets meeting only finitely many sets in
2793:
who don't, which means that they are contained in the closed set
1441:
Proof that paracompact
Hausdorff spaces admit partitions of unity
5419:, Progress in Mathematics, vol. 107, Springer, p. 32,
59:, and a Hausdorff space is paracompact if and only if it admits
1138:. (The long line is locally compact, but not second countable.)
5416:
Loop Spaces, Characteristic
Classes and Geometric Quantization
3865:{\displaystyle S=\{U\in {\mathcal {O}}:N{\text{ meets }}U\}\,}
1446:(Click "show" at right to see the proof or "hide" to hide it.)
371:{\displaystyle X\subseteq \bigcup _{\alpha \in A}U_{\alpha }.}
4748:
A product of paracompact spaces need not be paracompact. The
1845:
is a continuous function which is always non-zero and finite.
5088:{\displaystyle \left\{\alpha \in A:x\in U_{\alpha }\right\}}
4597:
4519:
4300:
4268:
4117:
4059:
3839:
3556:
3510:
3404:
2778:
2719:
2603:
2500:
2427:
2253:
2225:
2152:
2121:
2072:
2017:
1992:
1968:
1943:
1918:
1883:
1817:
1680:
1644:
1554:
1511:
is a locally finite open cover, then there exists open sets
1495:
1335:
subordinate to any open cover. This means the following: if
1118:
theory is not sufficient to prove it, even after the weaker
3934:, which is a continuous function; hence the preimage under
5119:
space is the same thing as a paracompact
Hausdorff space.
4726:
Every closed subset of a paracompact space is paracompact.
4355:
is continuous positive, finite-valued). So replacing each
1126:
Some examples of spaces that are not paracompact include:
938:
if every open cover has a locally finite open refinement.
1406:
and the sum of the nonzero functions is identically 1 in
4777:
if every open cover has an open point-finite refinement.
4655:{\displaystyle \operatorname {supp} ~f_{W}\subseteq W\,}
4246:{\displaystyle \operatorname {supp} ~f_{W}\subseteq W\,}
3371:{\displaystyle \operatorname {supp} ~f_{U}\subseteq U\,}
1784:{\displaystyle \operatorname {supp} ~f_{U}\subseteq U\,}
1433:
is first defined locally (where the manifold looks like
3328:{\displaystyle f_{U}\upharpoonright {\bar {W}}_{U}=1\,}
5046:
4870:
4678:
4625:
4587:
4547:
4515:
4494:
4473:
4446:
4425:
4389:
4361:
4332:
4290:
4259:
4216:
4164:
4087:
4055:
4022:
3991:
3961:
3940:
3878:
3821:
3794:
3770:
3740:
3697:
3676:
3642:
3621:
3578:
3553:
3532:
3507:
3486:
3465:
3438:
3384:
3341:
3285:
3233:
3200:
3148:
3128:
3108:
3088:
3068:
3048:
3006:
2980:
2938:
2840:
2799:
2769:
2740:
2678:
2651:
2624:
2611:{\displaystyle A_{1},...,A_{n},...\in {\mathcal {V}}}
2550:
2521:
2497:
2477:
2448:
2424:
2382:
2356:
2313:
2275:
2250:
2200:
2173:
2143:
2118:
2044:
2014:
1989:
1965:
1940:
1915:
1880:
1859:
1838:{\displaystyle f:=\sum _{U\in {\mathcal {O}}}f_{U}\,}
1797:
1754:
1702:
1677:
1619:
1576:
1545:
1517:
1492:
1456:
1281:
Paracompact spaces are sometimes required to also be
1234:
states that lower semicontinuous multifunctions from
1027:
1007:
987:
967:
947:
920:
856:
833:
813:
793:
773:
722:
690:
647:
627:
600:
577:
550:
496:
445:
421:
387:
332:
309:
289:
269:
214:
194:
170:
146:
3425:{\displaystyle f=\sum _{U\in {\mathcal {O}}}f_{U}\,}
439:
of some set in the old cover. In symbols, the cover
5472:(1944), "Une généralisation des espaces compacts",
5363:Good, C.; Tree, I. J.; Watson, W. S. (April 1998).
4158:. Applying Lemma 2, we obtain continuous functions
4081:a locally finite subcover of the refinement cover:
85:The notion of paracompact space is also studied in
5600:
5087:
4933:
4685:
4654:
4611:
4573:
4533:
4501:
4480:
4459:
4432:
4411:
4375:
4347:
4314:
4276:
4245:
4202:
4150:
4073:
4029:
3998:
3977:
3947:
3926:
3864:
3807:
3780:
3756:
3726:
3683:
3662:
3628:
3607:
3564:
3539:
3518:
3493:
3472:
3451:
3424:
3370:
3327:
3271:
3207:
3176:
3134:
3114:
3094:
3074:
3054:
3034:
2992:
2966:
2924:
2826:
2785:
2755:
2726:
2664:
2637:
2610:
2536:
2507:
2483:
2463:
2434:
2410:
2368:
2342:
2294:
2261:
2236:
2186:
2159:
2129:
2104:
2025:
2000:
1975:
1951:
1926:
1891:
1866:
1837:
1783:
1740:
1688:
1655:
1605:
1562:
1531:
1503:
1463:
1033:
1013:
993:
973:
953:
926:
903:
839:
819:
799:
779:
759:
696:
673:
633:
613:
583:
563:
533:
482:
427:
393:
370:
315:
295:
275:
251:
200:
176:
152:
89:, where it is more well-behaved. For example, the
4711:we end up with the compact spaces in both cases.
4706:There is a similarity between the definitions of
2727:{\displaystyle A_{1},...,A_{n}\in {\mathcal {V}}}
5370:Proceedings of the American Mathematical Society
5335:Proceedings of the American Mathematical Society
5330:"A new proof that metric spaces are paracompact"
5250:Proceedings of the American Mathematical Society
4326:function which is always finite non-zero (hence
4832:Definition of relevant terms for the variations
2442:to be locally finite, there is a neighbourhood
2237:{\displaystyle \{W_{U}:U\in {\mathcal {O}}\}\,}
1656:{\displaystyle \{W_{U}:U\in {\mathcal {O}}\}\,}
674:{\displaystyle V_{\beta }\subseteq U_{\alpha }}
760:{\displaystyle U=\{U_{\alpha }:\alpha \in A\}}
534:{\displaystyle U=\{U_{\alpha }:\alpha \in A\}}
252:{\displaystyle U=\{U_{\alpha }:\alpha \in A\}}
5203:Bulletin of the American Mathematical Society
1959:, and whose closure is contained in a set in
1269:which is used in the proof that a product of
78:. This is equivalent to requiring that every
8:
5475:Journal de Mathématiques Pures et Appliquées
5365:"On Stone's theorem and the axiom of choice"
5315:. Bull. Amer. Math. Soc. 54 (1948), 977–982
5099:As the names imply, a fully normal space is
4144:
4088:
3858:
3828:
2230:
2201:
2098:
2061:
1649:
1620:
754:
729:
528:
503:
483:{\displaystyle V=\{V_{\beta }:\beta \in B\}}
477:
452:
246:
221:
5133:gave a direct proof of the latter fact and
2343:{\displaystyle {\bar {W_{U}}}\subseteq U\,}
2302:, this cover is immediately locally finite.
1606:{\displaystyle {\bar {W_{U}}}\subseteq U\,}
1160:is not paracompact; in fact it is not even
1001:that intersect only finitely many opens in
5607:. Reading, Massachusetts: Addison-Wesley.
4574:{\displaystyle f_{W}\upharpoonright N=0\,}
1475:direction is straightforward. Now for the
1367:from the collection, there is an open set
1327:The most important feature of paracompact
767:is locally finite if and only if, for any
5382:
5347:
5262:
5074:
5045:
4922:
4904:
4899:
4877:
4872:
4869:
4718:Comparison of properties with compactness
4682:
4677:
4651:
4639:
4624:
4612:{\displaystyle W\in {\mathcal {O}}^{*}\,}
4608:
4602:
4596:
4595:
4586:
4570:
4552:
4546:
4530:
4524:
4518:
4517:
4514:
4498:
4493:
4477:
4472:
4456:
4445:
4429:
4424:
4408:
4400:
4394:
4388:
4372:
4366:
4360:
4344:
4336:
4331:
4315:{\displaystyle W\in {\mathcal {O}}^{*}\,}
4311:
4305:
4299:
4298:
4289:
4273:
4267:
4266:
4258:
4242:
4230:
4215:
4199:
4169:
4163:
4147:
4127:
4126:
4116:
4115:
4108:
4094:
4086:
4070:
4064:
4058:
4057:
4054:
4026:
4021:
3995:
3990:
3974:
3960:
3944:
3939:
3923:
3911:
3895:
3877:
3861:
3850:
3838:
3837:
3820:
3804:
3793:
3777:
3769:
3753:
3739:
3723:
3702:
3696:
3680:
3675:
3659:
3653:
3641:
3625:
3620:
3604:
3583:
3577:
3561:
3555:
3554:
3552:
3536:
3531:
3515:
3509:
3508:
3506:
3490:
3485:
3469:
3464:
3448:
3437:
3421:
3415:
3403:
3402:
3395:
3383:
3367:
3355:
3340:
3324:
3312:
3301:
3300:
3290:
3284:
3268:
3238:
3232:
3204:
3199:
3162:
3156:
3155:
3147:
3127:
3107:
3087:
3067:
3047:
3020:
3014:
3013:
3005:
2979:
2967:{\displaystyle {\bar {A_{i}}}\subseteq U}
2946:
2940:
2939:
2937:
2904:
2898:
2897:
2870:
2864:
2863:
2848:
2842:
2841:
2839:
2798:
2777:
2776:
2768:
2739:
2718:
2717:
2708:
2683:
2677:
2656:
2650:
2629:
2623:
2602:
2601:
2580:
2555:
2549:
2520:
2499:
2498:
2496:
2476:
2447:
2426:
2425:
2423:
2396:
2390:
2389:
2381:
2355:
2339:
2321:
2315:
2314:
2312:
2280:
2274:
2258:
2252:
2251:
2249:
2233:
2224:
2223:
2208:
2199:
2178:
2172:
2151:
2150:
2142:
2126:
2120:
2119:
2117:
2101:
2081:
2080:
2071:
2070:
2049:
2043:
2022:
2016:
2015:
2013:
1997:
1991:
1990:
1988:
1967:
1966:
1964:
1948:
1942:
1941:
1939:
1923:
1917:
1916:
1914:
1888:
1882:
1881:
1879:
1863:
1858:
1834:
1828:
1816:
1815:
1808:
1796:
1780:
1768:
1753:
1737:
1707:
1701:
1685:
1679:
1678:
1676:
1652:
1643:
1642:
1627:
1618:
1602:
1584:
1578:
1577:
1575:
1559:
1553:
1552:
1544:
1528:
1522:
1516:
1500:
1494:
1493:
1491:
1460:
1455:
1250:The product of a paracompact space and a
1026:
1006:
986:
966:
946:
919:
878:
855:
832:
812:
792:
772:
736:
721:
689:
665:
652:
646:
626:
605:
599:
576:
555:
549:
510:
495:
459:
444:
420:
386:
359:
343:
331:
308:
288:
268:
228:
213:
193:
169:
145:
44:
5400:
5227:
5181:
3082:is the complement of a neighbourhood of
1265:Both these results can be proved by the
5299:, preliminary version available on the
5169:
5162:
2809:
1479:direction, we do this in a few stages.
1303:Every paracompact Hausdorff space is a
1296:) Every paracompact Hausdorff space is
1106:. Existing proofs of this require the
981:is locally finite iff the set of opens
893:
116:if and only if it is a paracompact and
112:is paracompact. A topological space is
5445:Convergence and Uniformity in Topology
3547:belongs to only finitely many sets in
3177:{\displaystyle x\notin {\bar {W_{U}}}}
3035:{\displaystyle x\notin {\bar {A_{i}}}}
2411:{\displaystyle x\notin {\bar {W_{U}}}}
1130:The most famous counterexample is the
4277:{\displaystyle U\in {\mathcal {O}}\,}
2491:such that only finitely many sets in
1563:{\displaystyle U\in {\mathcal {O}}\,}
1064:is paracompact. In particular, every
7:
4665:
4534:{\displaystyle {\mathcal {O}}^{*}\,}
4322:; for which their sum constitutes a
4074:{\displaystyle {\mathcal {O}}^{*}\,}
4009:
3432:is always finite and non-zero, take
3187:
716:many sets in the cover. In symbols,
4741:It is different in these respects:
4509:meeting only finitely many sets in
3501:meeting only finitely many sets in
2786:{\displaystyle A\in {\mathcal {V}}}
2160:{\displaystyle A\in {\mathcal {V}}}
1261:and a compact space is metacompact.
1185:shows that there are 2 topological
5313:Paracompactness and product spaces
4105:
807:, there exists some neighbourhood
708:if every point of the space has a
51:is paracompact. Every paracompact
43:. These spaces were introduced by
14:
5349:10.1090/S0002-9939-1969-0236876-3
5264:10.1090/S0002-9939-1953-0056905-8
5196:"The point of pointless topology"
5137:gave another, elementary, proof.
2515:have non-empty intersection with
1853:In a paracompact Hausdorff space
1310:On paracompact Hausdorff spaces,
5676:Properties of topological spaces
5282:from the original on 2017-08-27.
4873:
4791:if every open cover has an open
4756:is a classical example for this.
3788:. To establish continuity, take
3565:{\displaystyle {\mathcal {O}}\,}
3519:{\displaystyle {\mathcal {O}}\,}
2295:{\displaystyle W_{U}\subseteq U}
2262:{\displaystyle {\mathcal {O}}\,}
2130:{\displaystyle {\mathcal {V}}\,}
2026:{\displaystyle {\mathcal {V}}\,}
2001:{\displaystyle {\mathcal {O}}\,}
1952:{\displaystyle {\mathcal {O}}\,}
1927:{\displaystyle {\mathcal {V}}\,}
1892:{\displaystyle {\mathcal {O}}\,}
1689:{\displaystyle {\mathcal {O}}\,}
1504:{\displaystyle {\mathcal {O}}\,}
1178:with no refinement of any kind.)
1156:. Any infinite set carrying the
5215:10.1090/S0273-0979-1983-15080-2
4837:Given a cover and a point, the
4686:{\displaystyle \blacksquare \,}
4030:{\displaystyle \blacksquare \,}
3208:{\displaystyle \blacksquare \,}
2827:{\displaystyle C:=X\setminus V}
2008:is locally finite. Now replace
1663:is a locally finite refinement.
914:is finite. A topological space
381:A cover of a topological space
5242:"A note on paracompact spaces"
4889:
4883:
4558:
4196:
4184:
4181:
4132:
4123:
4102:
3971:
3965:
3917:
3882:
3750:
3744:
3714:
3708:
3595:
3589:
3306:
3296:
3265:
3253:
3250:
3168:
3026:
2952:
2910:
2876:
2854:
2821:
2815:
2750:
2744:
2645:. Therefore we can decompose
2531:
2525:
2508:{\displaystyle {\mathcal {V}}}
2458:
2452:
2435:{\displaystyle {\mathcal {V}}}
2402:
2327:
2307:Now we want to show that each
2086:
1976:{\displaystyle {\mathcal {O}}}
1734:
1722:
1719:
1590:
1114:case. It has been shown that
1:
5384:10.1090/S0002-9939-98-04163-X
4702:Relationship with compactness
4203:{\displaystyle f_{W}:X\to \,}
3272:{\displaystyle f_{U}:X\to \,}
1741:{\displaystyle f_{U}:X\to \,}
1371:from the cover such that the
490:is a refinement of the cover
93:of any number of paracompact
5413:Brylinski, Jean-Luc (2007),
5403:, pp. 165, Theorem 2.2.
5194:Johnstone, Peter T. (1983).
5184:, pp. 170, Theorem 4.2.
3727:{\displaystyle f_{U}(x)=1\,}
3663:{\displaystyle x\in W_{U}\,}
3608:{\displaystyle f_{U}(x)=0\,}
1285:to extend their properties.
1277:Paracompact Hausdorff spaces
1189:of non-paracompact surfaces.
1141:Another counterexample is a
1134:, which is a nonparacompact
5644:Encyclopedia of Mathematics
5541:Counterexamples in Topology
5509:. Boston: Allyn and Bacon.
5297:Vector bundles and K-theory
5230:, pp. 165 Theorem 2.4.
5032:is point finite if for any
4754:in the lower limit topology
3985:will be a neighbourhood of
2618:those in the definition of
1273:compact spaces is compact.
1226:Smirnov metrization theorem
614:{\displaystyle U_{\alpha }}
5692:
5624:"Topology/Paracompactness"
4802:if it is fully normal and
4581:for all but finitely many
3615:for all but finitely many
1394:, there is a neighborhood
564:{\displaystyle V_{\beta }}
5671:Compactness (mathematics)
5599:Willard, Stephen (1970).
4412:{\displaystyle f_{W}/f\,}
2993:{\displaystyle x\notin U}
2369:{\displaystyle x\notin U}
2244:is an open refinement of
1232:Michael selection theorem
1158:particular point topology
1120:axiom of dependent choice
684:An open cover of a space
82:subspace be paracompact.
5240:Michael, Ernest (1953).
4964:is a star refinement of
4768:A topological space is:
4750:square of the real line
4460:{\displaystyle x\in X\,}
3872:, which is finite; then
3781:{\displaystyle \geq 1\,}
3452:{\displaystyle x\in X\,}
3279:be continuous maps with
101:, which states that the
76:hereditarily paracompact
4376:{\displaystyle f_{W}\,}
1532:{\displaystyle W_{U}\,}
407:if all its members are
5570:Upper Saddle River, NJ
5536:J. Arthur Seebach, Jr.
5089:
4956:of a cover of a space
4935:
4687:
4656:
4613:
4575:
4535:
4503:
4488:be a neighbourhood of
4482:
4461:
4434:
4413:
4377:
4349:
4316:
4278:
4247:
4204:
4152:
4075:
4031:
4000:
3979:
3978:{\displaystyle f(x)\,}
3955:of a neighbourhood of
3949:
3928:
3866:
3809:
3782:
3758:
3757:{\displaystyle f(x)\,}
3728:
3685:
3664:
3630:
3609:
3566:
3541:
3520:
3495:
3474:
3453:
3426:
3372:
3329:
3273:
3227:Applying Lemma 1, let
3209:
3178:
3136:
3116:
3096:
3076:
3056:
3036:
2994:
2968:
2926:
2828:
2787:
2757:
2728:
2666:
2639:
2612:
2538:
2509:
2485:
2465:
2436:
2412:
2370:
2344:
2296:
2263:
2238:
2188:
2161:
2137:guarantees that every
2131:
2106:
2027:
2002:
1977:
1953:
1928:
1893:
1868:
1839:
1785:
1742:
1690:
1657:
1607:
1564:
1533:
1505:
1465:
1072:second-countable space
1035:
1015:
995:
975:
955:
928:
905:
841:
821:
801:
781:
761:
698:
675:
635:
615:
585:
565:
535:
484:
429:
415:of a cover of a space
395:
372:
317:
297:
277:
253:
202:
178:
154:
5090:
4936:
4688:
4657:
4614:
4576:
4536:
4504:
4483:
4462:
4435:
4414:
4378:
4350:
4348:{\displaystyle 1/f\,}
4317:
4279:
4248:
4205:
4153:
4076:
4032:
4001:
3980:
3950:
3929:
3867:
3810:
3808:{\displaystyle x,N\,}
3783:
3759:
3729:
3686:
3665:
3631:
3610:
3567:
3542:
3521:
3496:
3475:
3454:
3427:
3373:
3330:
3274:
3210:
3179:
3137:
3117:
3097:
3077:
3057:
3037:
2995:
2969:
2927:
2829:
2788:
2758:
2729:
2667:
2665:{\displaystyle W_{U}}
2640:
2638:{\displaystyle W_{U}}
2613:
2539:
2510:
2486:
2466:
2437:
2413:
2376:, we will prove that
2371:
2345:
2297:
2264:
2239:
2189:
2187:{\displaystyle W_{U}}
2167:is contained in some
2162:
2132:
2107:
2028:
2003:
1978:
1954:
1929:
1894:
1869:
1840:
1786:
1743:
1691:
1658:
1608:
1565:
1534:
1506:
1466:
1104:M. E. Rudin
1036:
1021:also form a cover of
1016:
996:
976:
956:
929:
906:
842:
822:
802:
782:
762:
712:that intersects only
699:
676:
636:
616:
586:
566:
536:
485:
430:
396:
373:
318:
298:
278:
254:
203:
179:
155:
5044:
4868:
4676:
4623:
4585:
4545:
4513:
4492:
4471:
4444:
4423:
4387:
4359:
4330:
4288:
4257:
4214:
4162:
4085:
4053:
4020:
3989:
3959:
3938:
3876:
3819:
3792:
3768:
3738:
3695:
3674:
3640:
3619:
3576:
3551:
3530:
3505:
3484:
3463:
3436:
3382:
3339:
3283:
3231:
3198:
3146:
3142:. Therefore we have
3126:
3106:
3086:
3066:
3046:
3004:
2978:
2936:
2838:
2797:
2767:
2738:
2676:
2649:
2622:
2548:
2519:
2495:
2475:
2446:
2422:
2380:
2354:
2311:
2273:
2248:
2198:
2171:
2141:
2116:
2042:
2012:
1987:
1963:
1938:
1913:
1878:
1857:
1795:
1752:
1700:
1675:
1617:
1574:
1543:
1515:
1490:
1454:
1136:topological manifold
1025:
1005:
985:
965:
945:
918:
854:
831:
811:
791:
771:
720:
688:
645:
625:
598:
575:
548:
494:
443:
419:
385:
330:
307:
287:
267:
212:
192:
168:
144:
5639:"Paracompact space"
5568:(Second ed.).
5147:a-paracompact space
5016:A cover of a space
4502:{\displaystyle x\,}
4481:{\displaystyle N\,}
4433:{\displaystyle 1\,}
3999:{\displaystyle x\,}
3948:{\displaystyle f\,}
3815:as before, and let
3684:{\displaystyle U\,}
3629:{\displaystyle U\,}
3540:{\displaystyle x\,}
3494:{\displaystyle x\,}
3480:a neighbourhood of
3473:{\displaystyle N\,}
1867:{\displaystyle X\,}
1464:{\displaystyle X\,}
1355:for every function
1347:with values in the
1333:partitions of unity
1331:is that they admit
1323:Partitions of unity
1209:subspaces as well.
1187:equivalence classes
160:is a collection of
99:Tychonoff's theorem
61:partitions of unity
5574:Prentice Hall, Inc
5478:, Neuvième Série,
5085:
5006:) is contained in
4931:
4917:
4729:Every paracompact
4683:
4652:
4609:
4571:
4531:
4499:
4478:
4457:
4430:
4409:
4373:
4345:
4312:
4274:
4243:
4200:
4148:
4071:
4027:
3996:
3975:
3945:
3924:
3906:
3862:
3805:
3778:
3754:
3724:
3681:
3660:
3626:
3605:
3562:
3537:
3516:
3491:
3470:
3449:
3422:
3410:
3368:
3325:
3269:
3205:
3174:
3132:
3112:
3092:
3072:
3052:
3032:
2990:
2964:
2922:
2824:
2783:
2753:
2724:
2662:
2635:
2608:
2534:
2505:
2481:
2461:
2432:
2408:
2366:
2340:
2292:
2259:
2234:
2184:
2157:
2127:
2112:. The property of
2102:
2023:
1998:
1973:
1949:
1924:
1889:
1864:
1835:
1823:
1781:
1738:
1686:
1653:
1603:
1560:
1529:
1501:
1461:
1450:A Hausdorff space
1427:differential forms
1149:many copies of an
1031:
1011:
991:
971:
951:
934:is now said to be
924:
901:
847:such that the set
837:
817:
797:
777:
757:
694:
671:
631:
611:
581:
561:
531:
480:
425:
391:
368:
354:
313:
293:
273:
249:
198:
174:
150:
118:locally metrizable
87:pointless topology
5666:Separation axioms
5583:978-0-13-181629-9
5562:Munkres, James R.
5532:Lynn Arthur Steen
5516:978-0-697-06889-7
5328:(February 1969).
5326:Rudin, Mary Ellen
5301:author's homepage
4987:, there exists a
4895:
4811:separation axioms
4697:
4696:
4634:
4225:
4135:
4097:
4041:
4040:
3891:
3853:
3852: meets
3391:
3350:
3309:
3219:
3218:
3171:
3135:{\displaystyle C}
3115:{\displaystyle x}
3095:{\displaystyle x}
3075:{\displaystyle C}
3055:{\displaystyle i}
3029:
2955:
2913:
2879:
2857:
2756:{\displaystyle V}
2537:{\displaystyle V}
2484:{\displaystyle x}
2464:{\displaystyle V}
2418:. Since we chose
2405:
2330:
2089:
1804:
1763:
1593:
1570:, such that each
1259:metacompact space
1257:The product of a
1034:{\displaystyle X}
1014:{\displaystyle U}
994:{\displaystyle V}
974:{\displaystyle X}
954:{\displaystyle U}
927:{\displaystyle X}
840:{\displaystyle x}
820:{\displaystyle V}
800:{\displaystyle X}
780:{\displaystyle x}
697:{\displaystyle X}
634:{\displaystyle U}
593:there exists some
584:{\displaystyle V}
428:{\displaystyle X}
394:{\displaystyle X}
339:
316:{\displaystyle X}
296:{\displaystyle U}
276:{\displaystyle X}
208:. In symbols, if
201:{\displaystyle X}
177:{\displaystyle X}
153:{\displaystyle X}
29:topological space
25:paracompact space
5683:
5652:
5627:
5618:
5606:
5603:General Topology
5595:
5528:
5498:
5457:
5456:
5437:
5431:
5429:
5410:
5404:
5398:
5389:
5388:
5386:
5377:(4): 1211–1218.
5360:
5354:
5353:
5351:
5322:
5316:
5309:
5303:
5290:
5284:
5283:
5281:
5266:
5246:
5237:
5231:
5225:
5219:
5218:
5200:
5191:
5185:
5179:
5173:
5167:
5152:Paranormal space
5094:
5092:
5091:
5086:
5084:
5080:
5079:
5078:
4940:
4938:
4937:
4932:
4927:
4926:
4916:
4909:
4908:
4882:
4881:
4876:
4692:
4690:
4689:
4684:
4666:
4661:
4659:
4658:
4653:
4644:
4643:
4632:
4618:
4616:
4615:
4610:
4607:
4606:
4601:
4600:
4580:
4578:
4577:
4572:
4557:
4556:
4540:
4538:
4537:
4532:
4529:
4528:
4523:
4522:
4508:
4506:
4505:
4500:
4487:
4485:
4484:
4479:
4466:
4464:
4463:
4458:
4439:
4437:
4436:
4431:
4418:
4416:
4415:
4410:
4404:
4399:
4398:
4382:
4380:
4379:
4374:
4371:
4370:
4354:
4352:
4351:
4346:
4340:
4321:
4319:
4318:
4313:
4310:
4309:
4304:
4303:
4283:
4281:
4280:
4275:
4272:
4271:
4252:
4250:
4249:
4244:
4235:
4234:
4223:
4209:
4207:
4206:
4201:
4174:
4173:
4157:
4155:
4154:
4149:
4137:
4136:
4128:
4122:
4121:
4120:
4098:
4096: open
4095:
4080:
4078:
4077:
4072:
4069:
4068:
4063:
4062:
4045:Proof (Theorem):
4036:
4034:
4033:
4028:
4010:
4005:
4003:
4002:
3997:
3984:
3982:
3981:
3976:
3954:
3952:
3951:
3946:
3933:
3931:
3930:
3925:
3916:
3915:
3905:
3871:
3869:
3868:
3863:
3854:
3851:
3843:
3842:
3814:
3812:
3811:
3806:
3787:
3785:
3784:
3779:
3763:
3761:
3760:
3755:
3733:
3731:
3730:
3725:
3707:
3706:
3690:
3688:
3687:
3682:
3669:
3667:
3666:
3661:
3658:
3657:
3635:
3633:
3632:
3627:
3614:
3612:
3611:
3606:
3588:
3587:
3571:
3569:
3568:
3563:
3560:
3559:
3546:
3544:
3543:
3538:
3525:
3523:
3522:
3517:
3514:
3513:
3500:
3498:
3497:
3492:
3479:
3477:
3476:
3471:
3458:
3456:
3455:
3450:
3431:
3429:
3428:
3423:
3420:
3419:
3409:
3408:
3407:
3377:
3375:
3374:
3369:
3360:
3359:
3348:
3334:
3332:
3331:
3326:
3317:
3316:
3311:
3310:
3302:
3295:
3294:
3278:
3276:
3275:
3270:
3243:
3242:
3223:Proof (Lemma 2):
3214:
3212:
3211:
3206:
3188:
3183:
3181:
3180:
3175:
3173:
3172:
3167:
3166:
3157:
3141:
3139:
3138:
3133:
3121:
3119:
3118:
3113:
3101:
3099:
3098:
3093:
3081:
3079:
3078:
3073:
3061:
3059:
3058:
3053:
3041:
3039:
3038:
3033:
3031:
3030:
3025:
3024:
3015:
2999:
2997:
2996:
2991:
2973:
2971:
2970:
2965:
2957:
2956:
2951:
2950:
2941:
2931:
2929:
2928:
2923:
2915:
2914:
2909:
2908:
2899:
2881:
2880:
2875:
2874:
2865:
2859:
2858:
2853:
2852:
2843:
2833:
2831:
2830:
2825:
2792:
2790:
2789:
2784:
2782:
2781:
2762:
2760:
2759:
2754:
2733:
2731:
2730:
2725:
2723:
2722:
2713:
2712:
2688:
2687:
2671:
2669:
2668:
2663:
2661:
2660:
2644:
2642:
2641:
2636:
2634:
2633:
2617:
2615:
2614:
2609:
2607:
2606:
2585:
2584:
2560:
2559:
2543:
2541:
2540:
2535:
2514:
2512:
2511:
2506:
2504:
2503:
2490:
2488:
2487:
2482:
2470:
2468:
2467:
2462:
2441:
2439:
2438:
2433:
2431:
2430:
2417:
2415:
2414:
2409:
2407:
2406:
2401:
2400:
2391:
2375:
2373:
2372:
2367:
2349:
2347:
2346:
2341:
2332:
2331:
2326:
2325:
2316:
2301:
2299:
2298:
2293:
2285:
2284:
2269:. Since we have
2268:
2266:
2265:
2260:
2257:
2256:
2243:
2241:
2240:
2235:
2229:
2228:
2213:
2212:
2193:
2191:
2190:
2185:
2183:
2182:
2166:
2164:
2163:
2158:
2156:
2155:
2136:
2134:
2133:
2128:
2125:
2124:
2111:
2109:
2108:
2103:
2091:
2090:
2082:
2076:
2075:
2054:
2053:
2032:
2030:
2029:
2024:
2021:
2020:
2007:
2005:
2004:
1999:
1996:
1995:
1982:
1980:
1979:
1974:
1972:
1971:
1958:
1956:
1955:
1950:
1947:
1946:
1933:
1931:
1930:
1925:
1922:
1921:
1905:Proof (Lemma 1):
1898:
1896:
1895:
1890:
1887:
1886:
1873:
1871:
1870:
1865:
1844:
1842:
1841:
1836:
1833:
1832:
1822:
1821:
1820:
1790:
1788:
1787:
1782:
1773:
1772:
1761:
1747:
1745:
1744:
1739:
1712:
1711:
1695:
1693:
1692:
1687:
1684:
1683:
1662:
1660:
1659:
1654:
1648:
1647:
1632:
1631:
1612:
1610:
1609:
1604:
1595:
1594:
1589:
1588:
1579:
1569:
1567:
1566:
1561:
1558:
1557:
1538:
1536:
1535:
1530:
1527:
1526:
1510:
1508:
1507:
1502:
1499:
1498:
1470:
1468:
1467:
1462:
1386:for every point
1379:is contained in
1329:Hausdorff spaces
1312:sheaf cohomology
1194:Sorgenfrey plane
1040:
1038:
1037:
1032:
1020:
1018:
1017:
1012:
1000:
998:
997:
992:
980:
978:
977:
972:
960:
958:
957:
952:
933:
931:
930:
925:
910:
908:
907:
902:
900:
896:
883:
882:
846:
844:
843:
838:
826:
824:
823:
818:
806:
804:
803:
798:
786:
784:
783:
778:
766:
764:
763:
758:
741:
740:
703:
701:
700:
695:
680:
678:
677:
672:
670:
669:
657:
656:
640:
638:
637:
632:
620:
618:
617:
612:
610:
609:
590:
588:
587:
582:
570:
568:
567:
562:
560:
559:
541:if and only if,
540:
538:
537:
532:
515:
514:
489:
487:
486:
481:
464:
463:
434:
432:
431:
426:
400:
398:
397:
392:
377:
375:
374:
369:
364:
363:
353:
322:
320:
319:
314:
302:
300:
299:
294:
282:
280:
279:
274:
258:
256:
255:
250:
233:
232:
207:
205:
204:
199:
183:
181:
180:
175:
159:
157:
156:
151:
45:Dieudonné (1944)
5691:
5690:
5686:
5685:
5684:
5682:
5681:
5680:
5656:
5655:
5637:
5634:
5622:Mathew, Akhil.
5621:
5615:
5598:
5584:
5560:
5547:Springer Verlag
5517:
5503:Dugundji, James
5501:
5470:Dieudonné, Jean
5468:
5465:
5460:
5439:
5438:
5434:
5427:
5412:
5411:
5407:
5399:
5392:
5362:
5361:
5357:
5324:
5323:
5319:
5310:
5306:
5291:
5287:
5279:
5244:
5239:
5238:
5234:
5226:
5222:
5198:
5193:
5192:
5188:
5180:
5176:
5172:, pp. 252.
5168:
5164:
5160:
5143:
5118:
5114:
5111:. Every fully T
5110:
5106:
5070:
5051:
5047:
5042:
5041:
5012:
4993:
4974:
4953:star refinement
4918:
4900:
4871:
4866:
4865:
4855:
4834:
4807:
4800:
4793:star refinement
4763:
4731:Hausdorff space
4720:
4704:
4699:
4698:
4674:
4673:
4635:
4621:
4620:
4594:
4583:
4582:
4548:
4543:
4542:
4516:
4511:
4510:
4490:
4489:
4469:
4468:
4442:
4441:
4421:
4420:
4390:
4385:
4384:
4362:
4357:
4356:
4328:
4327:
4297:
4286:
4285:
4255:
4254:
4226:
4212:
4211:
4165:
4160:
4159:
4083:
4082:
4056:
4051:
4050:
4018:
4017:
3987:
3986:
3957:
3956:
3936:
3935:
3907:
3874:
3873:
3817:
3816:
3790:
3789:
3766:
3765:
3736:
3735:
3698:
3693:
3692:
3672:
3671:
3649:
3638:
3637:
3617:
3616:
3579:
3574:
3573:
3549:
3548:
3528:
3527:
3503:
3502:
3482:
3481:
3461:
3460:
3434:
3433:
3411:
3380:
3379:
3351:
3337:
3336:
3299:
3286:
3281:
3280:
3234:
3229:
3228:
3196:
3195:
3158:
3144:
3143:
3124:
3123:
3122:is also not in
3104:
3103:
3084:
3083:
3064:
3063:
3044:
3043:
3016:
3002:
3001:
2976:
2975:
2942:
2934:
2933:
2900:
2866:
2844:
2836:
2835:
2795:
2794:
2765:
2764:
2763:, and the rest
2736:
2735:
2734:who intersect
2704:
2679:
2674:
2673:
2652:
2647:
2646:
2625:
2620:
2619:
2576:
2551:
2546:
2545:
2517:
2516:
2493:
2492:
2473:
2472:
2444:
2443:
2420:
2419:
2392:
2378:
2377:
2352:
2351:
2317:
2309:
2308:
2276:
2271:
2270:
2246:
2245:
2204:
2196:
2195:
2174:
2169:
2168:
2139:
2138:
2114:
2113:
2045:
2040:
2039:
2010:
2009:
1985:
1984:
1961:
1960:
1936:
1935:
1911:
1910:
1876:
1875:
1855:
1854:
1824:
1793:
1792:
1764:
1750:
1749:
1703:
1698:
1697:
1673:
1672:
1623:
1615:
1614:
1580:
1572:
1571:
1541:
1540:
1518:
1513:
1512:
1488:
1487:
1452:
1451:
1447:
1443:
1435:Euclidean space
1429:on paracompact
1417:
1325:
1316:ÄŚech cohomology
1305:shrinking space
1279:
1254:is paracompact.
1242:is paracompact.
1221:is paracompact.
1203:
1183:bagpipe theorem
1169:PrĂĽfer manifold
1108:axiom of choice
1088:is paracompact.
1079:Sorgenfrey line
1074:is paracompact.
1066:locally compact
1054:is paracompact.
1047:
1023:
1022:
1003:
1002:
983:
982:
963:
962:
943:
942:
916:
915:
874:
861:
857:
852:
851:
829:
828:
809:
808:
789:
788:
769:
768:
732:
718:
717:
686:
685:
661:
648:
643:
642:
623:
622:
601:
596:
595:
573:
572:
551:
546:
545:
506:
492:
491:
455:
441:
440:
417:
416:
383:
382:
355:
328:
327:
305:
304:
285:
284:
265:
264:
224:
210:
209:
190:
189:
166:
165:
142:
141:
129:
121:Hausdorff space
53:Hausdorff space
31:in which every
17:
12:
11:
5:
5689:
5687:
5679:
5678:
5673:
5668:
5658:
5657:
5654:
5653:
5633:
5632:External links
5630:
5629:
5628:
5619:
5613:
5596:
5582:
5558:
5529:
5515:
5499:
5464:
5461:
5459:
5458:
5441:Tukey, John W.
5432:
5425:
5405:
5390:
5355:
5317:
5311:Stone, A. H.
5304:
5293:Hatcher, Allen
5285:
5257:(5): 831–838.
5232:
5220:
5186:
5174:
5161:
5159:
5156:
5155:
5154:
5149:
5142:
5139:
5131:Ernest Michael
5116:
5112:
5108:
5104:
5097:
5096:
5083:
5077:
5073:
5069:
5066:
5063:
5060:
5057:
5054:
5050:
5014:
5010:
4991:
4972:
4947:
4946:
4942:
4941:
4930:
4925:
4921:
4915:
4912:
4907:
4903:
4898:
4894:
4891:
4888:
4885:
4880:
4875:
4862:
4861:
4853:
4833:
4830:
4815:
4814:
4805:
4798:
4786:
4778:
4762:
4759:
4758:
4757:
4746:
4739:
4738:
4727:
4719:
4716:
4703:
4700:
4695:
4694:
4681:
4671:
4669:
4664:
4663:
4650:
4647:
4642:
4638:
4631:
4628:
4605:
4599:
4593:
4590:
4569:
4566:
4563:
4560:
4555:
4551:
4527:
4521:
4497:
4476:
4455:
4452:
4449:
4440:. Finally for
4428:
4407:
4403:
4397:
4393:
4369:
4365:
4343:
4339:
4335:
4308:
4302:
4296:
4293:
4270:
4265:
4262:
4241:
4238:
4233:
4229:
4222:
4219:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4172:
4168:
4146:
4143:
4140:
4134:
4131:
4125:
4119:
4114:
4111:
4107:
4104:
4101:
4093:
4090:
4067:
4061:
4047:
4039:
4038:
4025:
4015:
4013:
4008:
4007:
3994:
3973:
3970:
3967:
3964:
3943:
3922:
3919:
3914:
3910:
3904:
3901:
3898:
3894:
3890:
3887:
3884:
3881:
3860:
3857:
3849:
3846:
3841:
3836:
3833:
3830:
3827:
3824:
3803:
3800:
3797:
3776:
3773:
3764:is finite and
3752:
3749:
3746:
3743:
3722:
3719:
3716:
3713:
3710:
3705:
3701:
3679:
3656:
3652:
3648:
3645:
3624:
3603:
3600:
3597:
3594:
3591:
3586:
3582:
3558:
3535:
3512:
3489:
3468:
3447:
3444:
3441:
3418:
3414:
3406:
3401:
3398:
3394:
3390:
3387:
3366:
3363:
3358:
3354:
3347:
3344:
3323:
3320:
3315:
3308:
3305:
3298:
3293:
3289:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3241:
3237:
3225:
3217:
3216:
3203:
3193:
3191:
3186:
3185:
3170:
3165:
3161:
3154:
3151:
3131:
3111:
3091:
3071:
3051:
3028:
3023:
3019:
3012:
3009:
2989:
2986:
2983:
2963:
2960:
2954:
2949:
2945:
2921:
2918:
2912:
2907:
2903:
2896:
2893:
2890:
2887:
2884:
2878:
2873:
2869:
2862:
2856:
2851:
2847:
2834:. We now have
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2780:
2775:
2772:
2752:
2749:
2746:
2743:
2721:
2716:
2711:
2707:
2703:
2700:
2697:
2694:
2691:
2686:
2682:
2672:in two parts:
2659:
2655:
2632:
2628:
2605:
2600:
2597:
2594:
2591:
2588:
2583:
2579:
2575:
2572:
2569:
2566:
2563:
2558:
2554:
2544:, and we note
2533:
2530:
2527:
2524:
2502:
2480:
2460:
2457:
2454:
2451:
2429:
2404:
2399:
2395:
2388:
2385:
2365:
2362:
2359:
2338:
2335:
2329:
2324:
2320:
2305:
2303:
2291:
2288:
2283:
2279:
2255:
2232:
2227:
2222:
2219:
2216:
2211:
2207:
2203:
2181:
2177:
2154:
2149:
2146:
2123:
2100:
2097:
2094:
2088:
2085:
2079:
2074:
2069:
2066:
2063:
2060:
2057:
2052:
2048:
2038:Now we define
2035:
2034:
2019:
1994:
1970:
1945:
1920:
1907:
1901:
1900:
1885:
1862:
1847:
1846:
1831:
1827:
1819:
1814:
1811:
1807:
1803:
1800:
1791:and such that
1779:
1776:
1771:
1767:
1760:
1757:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1710:
1706:
1682:
1665:
1664:
1651:
1646:
1641:
1638:
1635:
1630:
1626:
1622:
1601:
1598:
1592:
1587:
1583:
1556:
1551:
1548:
1525:
1521:
1497:
1459:
1448:
1445:
1444:
1442:
1439:
1415:
1412:
1411:
1384:
1324:
1321:
1320:
1319:
1308:
1301:
1293:Jean Dieudonné
1278:
1275:
1263:
1262:
1255:
1244:
1243:
1229:
1222:
1219:Lindelöf space
1202:
1199:
1198:
1197:
1190:
1179:
1165:
1154:discrete space
1139:
1124:
1123:
1089:
1082:
1075:
1062:Lindelöf space
1055:
1046:
1043:
1030:
1010:
990:
970:
950:
923:
912:
911:
899:
895:
892:
889:
886:
881:
877:
873:
870:
867:
864:
860:
836:
816:
796:
776:
756:
753:
750:
747:
744:
739:
735:
731:
728:
725:
706:locally finite
693:
668:
664:
660:
655:
651:
630:
608:
604:
580:
558:
554:
530:
527:
524:
521:
518:
513:
509:
505:
502:
499:
479:
476:
473:
470:
467:
462:
458:
454:
451:
448:
424:
390:
379:
378:
367:
362:
358:
352:
349:
346:
342:
338:
335:
312:
303:is a cover of
292:
272:
263:of subsets of
261:indexed family
248:
245:
242:
239:
236:
231:
227:
223:
220:
217:
197:
173:
149:
128:
125:
41:locally finite
15:
13:
10:
9:
6:
4:
3:
2:
5688:
5677:
5674:
5672:
5669:
5667:
5664:
5663:
5661:
5650:
5646:
5645:
5640:
5636:
5635:
5631:
5625:
5620:
5616:
5614:0-486-43479-6
5610:
5605:
5604:
5597:
5593:
5589:
5585:
5579:
5575:
5571:
5567:
5563:
5559:
5556:
5555:3-540-90312-7
5552:
5548:
5544:
5542:
5537:
5533:
5530:
5526:
5522:
5518:
5512:
5508:
5504:
5500:
5497:
5493:
5489:
5485:
5481:
5477:
5476:
5471:
5467:
5466:
5462:
5454:
5450:
5446:
5442:
5436:
5433:
5428:
5426:9780817647308
5422:
5418:
5417:
5409:
5406:
5402:
5401:Dugundji 1966
5397:
5395:
5391:
5385:
5380:
5376:
5372:
5371:
5366:
5359:
5356:
5350:
5345:
5341:
5337:
5336:
5331:
5327:
5321:
5318:
5314:
5308:
5305:
5302:
5298:
5294:
5289:
5286:
5278:
5274:
5270:
5265:
5260:
5256:
5252:
5251:
5243:
5236:
5233:
5229:
5228:Dugundji 1966
5224:
5221:
5216:
5212:
5208:
5204:
5197:
5190:
5187:
5183:
5182:Dugundji 1966
5178:
5175:
5171:
5166:
5163:
5157:
5153:
5150:
5148:
5145:
5144:
5140:
5138:
5136:
5132:
5128:
5127:John W. Tukey
5123:
5120:
5103:and a fully T
5102:
5081:
5075:
5071:
5067:
5064:
5061:
5058:
5055:
5052:
5048:
5039:
5035:
5031:
5027:
5023:
5019:
5015:
5009:
5005:
5001:
4997:
4990:
4986:
4982:
4979:} if for any
4978:
4975: : α in
4971:
4967:
4963:
4959:
4955:
4954:
4949:
4948:
4944:
4943:
4928:
4923:
4919:
4913:
4910:
4905:
4901:
4896:
4892:
4886:
4878:
4864:
4863:
4859:
4856: : α in
4852:
4848:
4844:
4840:
4836:
4835:
4831:
4829:
4826:
4825:open covers.
4824:
4820:
4812:
4808:
4801:
4794:
4790:
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4779:
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1409:
1405:
1401:
1397:
1393:
1389:
1385:
1382:
1378:
1374:
1370:
1366:
1363: →
1362:
1358:
1354:
1353:
1352:
1350:
1349:unit interval
1346:
1343:functions on
1342:
1338:
1334:
1330:
1322:
1317:
1313:
1309:
1306:
1302:
1299:
1295:
1294:
1288:
1287:
1286:
1284:
1276:
1274:
1272:
1271:finitely many
1268:
1260:
1256:
1253:
1252:compact space
1249:
1248:
1247:
1241:
1237:
1233:
1230:
1227:
1223:
1220:
1216:
1215:regular space
1212:
1211:
1210:
1208:
1200:
1195:
1191:
1188:
1184:
1180:
1177:
1173:
1170:
1166:
1163:
1159:
1155:
1152:
1148:
1144:
1140:
1137:
1133:
1129:
1128:
1127:
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1096:
1090:
1087:
1083:
1080:
1076:
1073:
1070:
1067:
1063:
1060:
1056:
1053:
1052:compact space
1049:
1048:
1044:
1042:
1028:
1008:
988:
968:
948:
939:
937:
921:
897:
890:
887:
884:
879:
875:
871:
868:
865:
862:
858:
850:
849:
848:
834:
814:
794:
774:
751:
748:
745:
742:
737:
733:
726:
723:
715:
711:
707:
691:
682:
666:
662:
658:
653:
649:
628:
606:
602:
594:
578:
556:
552:
544:
525:
522:
519:
516:
511:
507:
500:
497:
474:
471:
468:
465:
460:
456:
449:
446:
438:
422:
414:
410:
406:
405:
388:
365:
360:
356:
350:
347:
344:
340:
336:
333:
326:
325:
324:
310:
290:
270:
262:
243:
240:
237:
234:
229:
225:
218:
215:
195:
187:
171:
163:
147:
140:
136:
135:
126:
124:
122:
119:
115:
111:
106:
104:
100:
96:
92:
88:
83:
81:
77:
72:
69:
64:
62:
58:
54:
50:
49:compact space
46:
42:
38:
34:
30:
26:
22:
5642:
5602:
5565:
5539:
5506:
5479:
5473:
5444:
5435:
5415:
5408:
5374:
5368:
5358:
5339:
5333:
5320:
5307:
5296:
5288:
5254:
5248:
5235:
5223:
5209:(1): 41–53.
5206:
5202:
5189:
5177:
5170:Munkres 2000
5165:
5124:
5121:
5098:
5037:
5033:
5029:
5026:point finite
5025:
5022:point-finite
5021:
5017:
5007:
5003:
4999:
4995:
4988:
4984:
4980:
4976:
4969:
4965:
4961:
4957:
4951:
4857:
4850:
4846:
4842:
4838:
4827:
4818:
4817:The adverb "
4816:
4796:
4789:fully normal
4788:
4782:orthocompact
4780:
4772:
4767:
4764:
4751:
4740:
4721:
4713:
4705:
4323:
4044:
3222:
3062:. And since
2350:. For every
2194:. Therefore
1904:
1850:
1668:
1483:
1476:
1472:
1449:
1424:
1414:In fact, a T
1413:
1407:
1403:
1399:
1395:
1391:
1387:
1380:
1376:
1368:
1364:
1360:
1356:
1344:
1336:
1326:
1290:
1280:
1270:
1264:
1245:
1239:
1235:
1225:
1204:
1175:
1171:
1125:
1110:for the non-
1100:metric space
1092:
940:
935:
913:
710:neighborhood
705:
683:
412:
402:
380:
132:
130:
110:metric space
107:
84:
75:
65:
35:has an open
24:
18:
4774:metacompact
4708:compactness
4619:since each
4284:, for each
3636:; moreover
1351:such that:
1291:Theorem of
1162:metacompact
1147:uncountably
1095:A. H. Stone
1093:Theorem of
936:paracompact
21:mathematics
5660:Categories
5463:References
5342:(2): 603.
5135:M.E. Rudin
5107:space is T
5095:is finite.
5040:, the set
4998:such that
4761:Variations
4541:, we have
4467:, letting
4324:continuous
3459:, and let
3042:for every
3000:, we have
1748:such that
1341:continuous
1318:are equal.
1267:tube lemma
1201:Properties
1086:CW complex
641:such that
413:refinement
127:Definition
114:metrizable
37:refinement
33:open cover
5649:EMS Press
5525:395340485
5488:0021-7824
5482:: 65–76,
5273:0002-9939
5076:α
5068:∈
5056:∈
5053:α
4924:α
4911:∋
4906:α
4897:⋃
4879:∗
4823:countable
4819:countably
4680:◼
4646:⊆
4630:
4604:∗
4592:∈
4559:↾
4526:∗
4451:∈
4307:∗
4295:∈
4264:∈
4237:⊆
4221:
4182:→
4139:⊆
4133:¯
4113:∈
4106:∃
4066:∗
4024:◼
3918:↾
3900:∈
3893:∑
3883:↾
3835:∈
3772:≥
3670:for some
3647:∈
3443:∈
3400:∈
3393:∑
3362:⊆
3346:
3307:¯
3297:↾
3251:→
3202:◼
3169:¯
3153:∉
3027:¯
3011:∉
2985:∉
2959:⊆
2953:¯
2917:∪
2911:¯
2895:∪
2883:∪
2877:¯
2861:⊆
2855:¯
2810:∖
2774:∈
2715:∈
2599:∈
2403:¯
2387:∉
2361:∉
2334:⊆
2328:¯
2287:⊆
2221:∈
2148:∈
2093:⊆
2087:¯
2068:∈
2059:⋃
1813:∈
1806:∑
1775:⊆
1759:
1720:→
1640:∈
1597:⊆
1591:¯
1550:∈
1539:for each
1431:manifolds
1283:Hausdorff
1132:long line
1122:is added.
1112:separable
1069:Hausdorff
894:∅
891:≠
885:∩
880:α
866:∈
863:α
749:∈
746:α
738:α
667:α
659:⊆
654:β
607:α
557:β
543:for every
523:∈
520:α
512:α
472:∈
469:β
461:β
409:open sets
361:α
348:∈
345:α
341:⋃
337:⊆
241:∈
238:α
230:α
188:contains
5592:42683260
5566:Topology
5564:(2000).
5557:. P.23.
5549:, 1978,
5507:Topology
5505:(1966).
5443:(1940).
5277:Archived
5141:See also
2932:. Since
1851:Theorem:
1669:Lemma 2:
1484:Lemma 1:
1151:infinite
1098:) Every
1045:Examples
714:finitely
71:subspace
47:. Every
39:that is
5651:, 2001
5496:0013297
5453:0002515
4797:fully T
4037:(Lem 2)
3691:, thus
3572:; thus
3526:; thus
3215:(Lem 1)
1477:only if
1373:support
1359::
1207:F-sigma
1143:product
1059:regular
283:, then
162:subsets
103:product
95:locales
91:product
5611:
5590:
5580:
5553:
5543:(2 ed)
5523:
5513:
5494:
5486:
5451:
5423:
5271:
5101:normal
4795:, and
4735:normal
4633:
4224:
3349:
1762:
1298:normal
1084:Every
1057:Every
1050:Every
437:subset
259:is an
184:whose
108:Every
68:closed
66:Every
57:normal
5280:(PDF)
5245:(PDF)
5199:(PDF)
5158:Notes
4809:(see
4693:(Thm)
4210:with
4049:Take
3734:; so
1874:, if
1420:below
186:union
137:of a
134:cover
27:is a
5609:ISBN
5588:OCLC
5578:ISBN
5551:ISBN
5534:and
5521:OCLC
5511:ISBN
5484:ISSN
5421:ISBN
5269:ISSN
5024:(or
4860:} is
4839:star
4627:supp
4218:supp
3343:supp
3335:and
2974:and
1909:Let
1756:supp
1613:and
1314:and
1192:The
1181:The
1167:The
1077:The
411:. A
404:open
80:open
23:, a
5379:doi
5375:126
5344:doi
5259:doi
5211:doi
5036:in
5020:is
4994:in
4983:in
4968:= {
4849:= {
4845:in
4733:is
4383:by
2471:of
1671:If
1486:If
1398:of
1390:in
1375:of
1145:of
961:of
827:of
787:in
704:is
621:in
571:in
401:is
323:if
164:of
139:set
55:is
19:In
5662::
5647:,
5641:,
5586:.
5576:.
5572::
5545:,
5538:,
5519:.
5492:MR
5490:,
5480:23
5449:MR
5393:^
5373:.
5367:.
5340:20
5338:.
5332:.
5295:,
5275:.
5267:.
5253:.
5247:.
5205:.
5201:.
4950:A
4893::=
4813:).
3102:,
2804::=
1802::=
1473:if
1213:A
1116:ZF
681:.
591:,
131:A
123:.
5626:.
5617:.
5594:.
5527:.
5455:.
5430:.
5387:.
5381::
5352:.
5346::
5261::
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5217:.
5213::
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5082:}
5072:U
5065:x
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5059:A
5049:{
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5013:.
5011:α
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5002:(
5000:V
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4929:.
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4887:x
4884:(
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4854:α
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4806:1
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4752:R
4737:.
4649:W
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4637:f
4598:O
4589:W
4568:0
4565:=
4562:N
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4550:f
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4475:N
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4402:/
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4185:[
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4176::
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4103:(
4100::
4092:V
4089:{
4060:O
4006:.
3993:x
3972:)
3969:x
3966:(
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3897:U
3889:=
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3880:f
3859:}
3856:U
3848:N
3845::
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3829:{
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3802:N
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3718:=
3715:)
3712:x
3709:(
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3602:0
3599:=
3596:)
3593:x
3590:(
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3446:X
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3389:=
3386:f
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3319:=
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3266:]
3263:1
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3257:0
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2902:A
2892:.
2889:.
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2751:]
2748:x
2745:[
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2710:n
2706:A
2702:,
2699:.
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2596:.
2593:.
2590:.
2587:,
2582:n
2578:A
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2571:.
2568:.
2565:.
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2532:]
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2526:[
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2453:[
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2231:}
2226:O
2218:U
2215::
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2202:{
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2078::
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2065:A
2062:{
2056:=
2051:U
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2018:V
1993:O
1969:O
1944:O
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1289:(
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1224:(
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1164:.
1091:(
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898:}
888:V
876:U
872::
869:A
859:{
835:x
815:V
795:X
775:x
755:}
752:A
743::
734:U
730:{
727:=
724:U
692:X
663:U
650:V
629:U
603:U
579:V
553:V
529:}
526:A
517::
508:U
504:{
501:=
498:U
478:}
475:B
466::
457:V
453:{
450:=
447:V
423:X
389:X
366:.
357:U
351:A
334:X
311:X
291:U
271:X
247:}
244:A
235::
226:U
222:{
219:=
216:U
196:X
172:X
148:X
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