403:
1411:, a product of a normal space and need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the
988:
is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the
951:" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T
1355:
864:
819:
1477:
if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.
766:
721:
644:
624:
600:
580:
560:
676:
1362:
with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.
955:
space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal
Hausdorff" instead of "T
1415:. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (
1789:
1597:
132:
114:
17:
1770:
1693:
1667:
1574:
1039:
414:, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods
1648:
1137:
603:
941:
246:
1168:
1164:
Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.
1334:
824:
779:
1497:
970:
1060:
are perfectly normal
Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
1794:
1503:
1480:
Counterexamples to some variations on these statements can be found in the lists above. Specifically,
1654:
Kemoto, Nobuyuki (2004). "Higher
Separation Axioms". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.).
1264:
1122:
369:
1257:
940:
may be.) The definitions given here are the ones usually used today. For more on this issue, see
773:
726:
681:
1584:
1416:
1175:
1126:
1075:
1025:
1002:
1256:. In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also
1057:
985:
8:
1451:
1141:
1068:
899:
891:
473:
1481:
1091:
1474:
1439:
1378:
1185:
1115:
1111:
1087:
1021:
966:
769:
629:
609:
585:
565:
545:
45:
1759:
649:
1766:
1689:
1663:
1644:
1636:
1400:
1107:
344:
292:
270:
52:
31:
1748:
1731:
1721:
1704:
1392:
In fact, any space that satisfies any one of these three conditions must be normal.
1079:
898:. Every perfectly normal space is completely normal, because perfect normality is a
1743:
1716:
1659:
1404:
1359:
1145:
1103:
1014:
308:
60:
40:
1546:
1463:
1412:
1386:
1167:
The main significance of normal spaces lies in the fact that they admit "enough"
1043:
990:
978:
450:
296:
168:
96:
1064:
928:" and derived concepts occasionally have a different meaning. (Nonetheless, "T
1783:
1685:
1677:
1396:
1253:
1197:
1153:
1032:
1010:
948:
485:
396:
355:
150:
1408:
1399:
of normal spaces is not necessarily normal. This fact was first proved by
1053:
Hausdorff spaces are normal, and all paracompact regular spaces are normal;
1171:
1149:
1050:
875:
262:
1547:"Why are these two definitions of a perfectly normal space equivalent?"
1370:
358:
284:
1532:
Engelking, Theorem 1.5.19. This is stated under the assumption of a T
1446:" to "normal completely regular" is the same as what we usually call
1130:
1178:, as expressed by the following theorems valid for any normal space
1455:
1432:
887:
435:
387:
that are also disjoint. More intuitively, this condition says that
258:
78:
1028:) are perfectly normal regular, although not in general Hausdorff;
894:. The equivalence between these three characterizations is called
1005:
are normal
Hausdorff spaces, or at least normal regular spaces:
768:. This is a stronger separation property than normality, as by
402:
1506: – Property of topological spaces stronger than normality
1500: – Property of topological spaces stronger than normality
1102:
An important example of a non-normal topology is given by the
1484:
is normal but not regular, while the space of functions from
422:, here represented by larger, but still disjoint, open disks.
1614:
866:, but not precisely separated in general. It turns out that
1426:
886:
is perfectly normal if and only if every closed set is the
1536:
space, but the proof does not make use of that assumption.
492:
is completely normal if and only if every open subset of
1121:
A non-normal space of some relevance to analysis is the
1337:
1094:
is an example of a normal space that is not regular.
827:
782:
729:
684:
652:
632:
612:
588:
568:
548:
606:, in the sense that there is a continuous function
1758:
1705:"On the topological product of paracompact spaces"
1385:. This shows the relationship of normal spaces to
1349:
858:
813:
760:
715:
670:
638:
618:
594:
574:
554:
959:", or "completely normal Hausdorff" instead of "T
1781:
1097:
1423:axiom are preserved under arbitrary products.
1407:. In fact, since there exist spaces which are
772:disjoint closed sets in a normal space can be
527:is Hausdorff; equivalently, every subspace of
484:is completely normal if and only if every two
977:are discussed elsewhere; they are related to
996:
1562:
488:can be separated by neighbourhoods. Also,
1702:
1287:, then there exists a continuous function
1204:, then there exists a continuous function
1747:
1720:
924:Note that the terms "normal space" and "T
921:, is a perfectly normal Hausdorff space.
1427:Relationships to other separation axioms
1078:is completely normal, and every regular
932:" always means the same as "completely T
562:in which every two disjoint closed sets
401:
311:and their further strengthenings define
1756:
1676:
1620:
1580:
1568:
1488:to itself is Tychonoff but not normal.
1403:. An example of this phenomenon is the
1350:{\displaystyle \emptyset \rightarrow X}
14:
1782:
1653:
1071:are hereditarily normal and Hausdorff.
496:is normal with the subspace topology.
445:that is normal; this is equivalent to
1729:
1248:. In fact, we can take the values of
1049:Generalizing the above examples, all
480:is a normal space. It turns out that
1732:"Paracompactness and product spaces"
874:is normal and every closed set is a
859:{\displaystyle F\subseteq f^{-1}(1)}
814:{\displaystyle E\subseteq f^{-1}(0)}
870:is perfectly normal if and only if
307:. These conditions are examples of
24:
1643:, Heldermann Verlag Berlin, 1989.
1473:A topological space is said to be
1338:
1090:are normal (even if not regular).
313:completely normal Hausdorff spaces
25:
1806:
1466:. These are what we usually call
1138:topology of pointwise convergence
1017:) are perfectly normal Hausdorff;
604:precisely separated by a function
325:perfectly normal Hausdorff spaces
1790:Properties of topological spaces
1656:Encyclopedia of General Topology
942:History of the separation axioms
1765:. Reading, MA: Addison-Wesley.
1749:10.1090/S0002-9904-1948-09118-2
1722:10.1090/S0002-9904-1947-08858-3
1559:Engelking, Theorem 2.1.6, p. 68
1442:. Thus, anything from "normal R
1156:metric spaces is never normal.
1140:. More generally, a theorem of
1590:
1553:
1539:
1526:
1517:
1341:
1279:is a continuous function from
853:
847:
808:
802:
749:
743:
704:
698:
665:
653:
338:
13:
1:
1630:
1159:
1098:Examples of non-normal spaces
30:For normal vector space, see
1510:
1035:Hausdorff spaces are normal;
936:", whatever the meaning of T
7:
1598:"separation axioms in nLab"
1498:Collectionwise normal space
1491:
1040:Stone–Čech compactification
1001:Most spaces encountered in
761:{\displaystyle f^{-1}(1)=F}
716:{\displaystyle f^{-1}(0)=E}
397:separated by neighbourhoods
10:
1811:
1504:Monotonically normal space
1252:to be entirely within the
515:, is a completely normal T
29:
1757:Willard, Stephen (1970).
1703:Sorgenfrey, R.H. (1947).
1454:, we see that all normal
1381:precisely subordinate to
997:Examples of normal spaces
468:, is a topological space
464:hereditarily normal space
242:
222:
202:
184:
167:
149:
131:
113:
95:
77:
59:
51:
39:
27:Type of topological space
1265:Tietze extension theorem
1123:topological vector space
261:and related branches of
1258:separated by a function
1026:pseudometrisable spaces
774:separated by a function
542:is a topological space
458:completely normal space
18:Completely normal space
1351:
1271:is a closed subset of
1076:second-countable space
860:
815:
762:
717:
672:
640:
620:
596:
576:
556:
540:perfectly normal space
423:
145:(completely Hausdorff)
1736:Bull. Amer. Math. Soc
1730:Stone, A. H. (1948).
1709:Bull. Amer. Math. Soc
1523:Willard, Exercise 15C
1438:, then it is in fact
1431:If a normal space is
1352:
1058:topological manifolds
1003:mathematical analysis
896:Vedenissoff's theorem
861:
816:
763:
718:
673:
641:
621:
597:
577:
557:
523:, which implies that
405:
283:: every two disjoint
1452:Kolmogorov quotients
1369:is a locally finite
1335:
1263:More generally, the
1136:to itself, with the
1069:totally ordered sets
1046:is normal Hausdorff;
986:locally normal space
825:
780:
727:
682:
650:
630:
610:
586:
566:
546:
1417:Tychonoff's theorem
1142:Arthur Harold Stone
1114:, which is used in
1088:fully normal spaces
1038:In particular, the
1022:pseudometric spaces
967:Fully normal spaces
947:Terms like "normal
900:hereditary property
892:continuous function
163:(regular Hausdorff)
1637:Engelking, Ryszard
1440:completely regular
1379:partition of unity
1377:, then there is a
1373:of a normal space
1347:
1303:in the sense that
1200:closed subsets of
1116:algebraic geometry
1112:spectrum of a ring
856:
811:
776:, in the sense of
758:
713:
668:
636:
616:
592:
572:
552:
424:
293:open neighborhoods
216:(completely normal
198:(normal Hausdorff)
46:topological spaces
1795:Separation axioms
1772:978-0-486-43479-7
1695:978-0-13-181629-9
1678:Munkres, James R.
1669:978-0-444-50355-8
1401:Robert Sorgenfrey
1212:to the real line
1108:algebraic variety
1015:metrizable spaces
639:{\displaystyle X}
619:{\displaystyle f}
595:{\displaystyle F}
575:{\displaystyle E}
555:{\displaystyle X}
449:being normal and
345:topological space
309:separation axioms
299:is also called a
271:topological space
255:
254:
236:(perfectly normal
41:Separation axioms
32:normal (geometry)
16:(Redirected from
1802:
1776:
1764:
1761:General Topology
1753:
1751:
1726:
1724:
1699:
1684:(2nd ed.).
1673:
1660:Elsevier Science
1641:General Topology
1624:
1618:
1612:
1611:
1609:
1608:
1594:
1588:
1578:
1572:
1566:
1560:
1557:
1551:
1550:
1543:
1537:
1530:
1524:
1521:
1482:Sierpiński space
1468:normal Hausdorff
1405:Sorgenfrey plane
1360:lifting property
1356:
1354:
1353:
1348:
1150:uncountably many
1144:states that the
1104:Zariski topology
1092:Sierpiński space
1065:order topologies
1056:All paracompact
882:. Equivalently,
865:
863:
862:
857:
846:
845:
820:
818:
817:
812:
801:
800:
767:
765:
764:
759:
742:
741:
722:
720:
719:
714:
697:
696:
677:
675:
674:
671:{\displaystyle }
669:
646:to the interval
645:
643:
642:
637:
625:
623:
622:
617:
601:
599:
598:
593:
581:
579:
578:
573:
561:
559:
558:
553:
472:such that every
466:
465:
406:The closed sets
238: Hausdorff)
233:
228:
218: Hausdorff)
213:
208:
195:
190:
175:
174:
160:
155:
142:
137:
122:
121:
106:
101:
88:
83:
70:
65:
37:
36:
21:
1810:
1809:
1805:
1804:
1803:
1801:
1800:
1799:
1780:
1779:
1773:
1742:(10): 977–982.
1696:
1670:
1633:
1628:
1627:
1619:
1615:
1606:
1604:
1596:
1595:
1591:
1579:
1575:
1567:
1563:
1558:
1554:
1545:
1544:
1540:
1535:
1531:
1527:
1522:
1518:
1513:
1494:
1459:
1445:
1436:
1429:
1422:
1413:Tychonoff plank
1387:paracompactness
1336:
1333:
1332:
1186:Urysohn's lemma
1162:
1100:
1044:Tychonoff space
1024:(and hence all
1013:(and hence all
999:
991:Nemytskii plane
979:paracompactness
974:
962:
958:
954:
939:
935:
931:
927:
918:
910:
879:
838:
834:
826:
823:
822:
793:
789:
781:
778:
777:
770:Urysohn's lemma
734:
730:
728:
725:
724:
689:
685:
683:
680:
679:
651:
648:
647:
631:
628:
627:
611:
608:
607:
587:
584:
583:
567:
564:
563:
547:
544:
543:
534:
518:
512:
504:
463:
462:
439:
431:
341:
332:
320:
304:
297:Hausdorff space
281:
276:that satisfies
251:
237:
231:
229:
226:
217:
211:
209:
206:
193:
191:
188:
176:
172:
171:
158:
156:
153:
140:
138:
135:
123:
119:
117:
104:
102:
99:
86:
84:
81:
68:
66:
63:
43:
35:
28:
23:
22:
15:
12:
11:
5:
1808:
1798:
1797:
1792:
1778:
1777:
1771:
1754:
1727:
1715:(6): 631–632.
1700:
1694:
1674:
1668:
1651:
1632:
1629:
1626:
1625:
1613:
1589:
1573:
1561:
1552:
1538:
1533:
1525:
1515:
1514:
1512:
1509:
1508:
1507:
1501:
1493:
1490:
1457:
1448:normal regular
1443:
1434:
1428:
1425:
1420:
1346:
1343:
1340:
1240:) = 1 for all
1224:) = 0 for all
1161:
1158:
1099:
1096:
1084:
1083:
1080:Lindelöf space
1074:Every regular
1072:
1061:
1054:
1047:
1036:
1029:
1018:
998:
995:
972:
960:
956:
952:
937:
933:
929:
925:
916:
908:
877:
855:
852:
849:
844:
841:
837:
833:
830:
810:
807:
804:
799:
796:
792:
788:
785:
757:
754:
751:
748:
745:
740:
737:
733:
712:
709:
706:
703:
700:
695:
692:
688:
667:
664:
661:
658:
655:
635:
615:
591:
571:
551:
532:
516:
510:
502:
486:separated sets
437:
429:
370:neighbourhoods
354:if, given any
340:
337:
330:
318:
302:
291:have disjoint
279:
253:
252:
250:
249:
243:
240:
239:
234:
225:
220:
219:
214:
205:
200:
199:
196:
187:
182:
181:
178:
170:
165:
164:
161:
152:
147:
146:
143:
134:
129:
128:
125:
116:
111:
110:
107:
98:
93:
92:
89:
80:
75:
74:
71:
62:
57:
56:
55:classification
49:
48:
26:
9:
6:
4:
3:
2:
1807:
1796:
1793:
1791:
1788:
1787:
1785:
1774:
1768:
1763:
1762:
1755:
1750:
1745:
1741:
1737:
1733:
1728:
1723:
1718:
1714:
1710:
1706:
1701:
1697:
1691:
1687:
1686:Prentice-Hall
1683:
1679:
1675:
1671:
1665:
1661:
1658:. Amsterdam:
1657:
1652:
1650:
1649:3-88538-006-4
1646:
1642:
1638:
1635:
1634:
1623:, Section 17.
1622:
1617:
1603:
1599:
1593:
1586:
1582:
1577:
1571:, p. 213
1570:
1565:
1556:
1548:
1542:
1529:
1520:
1516:
1505:
1502:
1499:
1496:
1495:
1489:
1487:
1483:
1478:
1476:
1471:
1469:
1465:
1461:
1453:
1449:
1441:
1437:
1424:
1418:
1414:
1410:
1406:
1402:
1398:
1393:
1390:
1388:
1384:
1380:
1376:
1372:
1368:
1363:
1361:
1357:
1344:
1328:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1299:that extends
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1261:
1259:
1255:
1254:unit interval
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1181:
1177:
1173:
1170:
1165:
1157:
1155:
1151:
1147:
1143:
1139:
1135:
1132:
1128:
1124:
1119:
1117:
1113:
1109:
1105:
1095:
1093:
1089:
1081:
1077:
1073:
1070:
1066:
1062:
1059:
1055:
1052:
1048:
1045:
1041:
1037:
1034:
1030:
1027:
1023:
1019:
1016:
1012:
1011:metric spaces
1008:
1007:
1006:
1004:
994:
992:
987:
982:
980:
976:
968:
964:
950:
949:regular space
945:
943:
922:
920:
912:
903:
901:
897:
893:
889:
885:
881:
873:
869:
850:
842:
839:
835:
831:
828:
805:
797:
794:
790:
786:
783:
775:
771:
755:
752:
746:
738:
735:
731:
710:
707:
701:
693:
690:
686:
662:
659:
656:
633:
613:
605:
589:
569:
549:
541:
536:
530:
526:
522:
514:
506:
497:
495:
491:
487:
483:
479:
475:
471:
467:
459:
454:
452:
448:
444:
441:
433:
421:
417:
413:
409:
404:
400:
398:
394:
390:
386:
382:
378:
374:
371:
367:
363:
360:
357:
353:
349:
346:
336:
334:
326:
322:
314:
310:
306:
298:
294:
290:
286:
282:
275:
272:
268:
264:
260:
248:
245:
244:
241:
235:
230:
221:
215:
210:
201:
197:
192:
183:
179:
177:
166:
162:
157:
148:
144:
139:
130:
126:
124:
112:
108:
103:
94:
90:
85:
76:
72:
67:
58:
54:
50:
47:
42:
38:
33:
19:
1760:
1739:
1735:
1712:
1708:
1681:
1655:
1640:
1621:Willard 1970
1616:
1605:. Retrieved
1601:
1592:
1581:Willard 1970
1576:
1569:Munkres 2000
1564:
1555:
1541:
1528:
1519:
1485:
1479:
1475:pseudonormal
1472:
1467:
1447:
1430:
1394:
1391:
1382:
1374:
1366:
1364:
1331:
1329:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1262:
1249:
1245:
1241:
1237:
1233:
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528:
524:
520:
509:completely T
508:
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365:
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351:
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312:
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277:
273:
267:normal space
266:
256:
223:
203:
185:
133:completely T
73:(Kolmogorov)
1602:ncatlab.org
1583:, pp.
1419:) and the T
1051:paracompact
915:perfectly T
531:must be a T
359:closed sets
339:Definitions
295:. A normal
285:closed sets
263:mathematics
180:(Tychonoff)
109:(Hausdorff)
1784:Categories
1631:References
1607:2021-10-12
1371:open cover
1319:) for all
1216:such that
1169:continuous
1160:Properties
1110:or on the
1086:Also, all
1082:is normal.
678:such that
53:Kolmogorov
1511:Citations
1464:Tychonoff
1450:. Taking
1342:→
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1176:functions
1131:real line
1129:from the
1127:functions
840:−
832:⊆
795:−
787:⊆
736:−
691:−
451:Hausdorff
127:(Urysohn)
91:(Fréchet)
1682:Topology
1680:(2000).
1492:See also
1470:spaces.
1358:has the
1330:The map
1198:disjoint
1196:are two
1174:-valued
888:zero set
474:subspace
356:disjoint
259:topology
1585:100–101
1397:product
1154:compact
1146:product
1125:of all
1033:compact
971:fully T
602:can be
535:space.
395:can be
278:Axiom T
247:History
173:3½
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1460:spaces
1409:Dowker
1106:on an
975:spaces
519:space
333:spaces
323:, and
321:spaces
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120:½
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911:space
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626:from
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505:space
460:, or
440:space
434:is a
432:space
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1767:ISBN
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1063:All
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154:3
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34:.
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