403:
1407:, 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
984:
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
947:" 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
1351:
864:
819:
1473:
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:
1358:
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.
951:
space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal
Hausdorff" instead of "T
1785:
1411:. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (
132:
114:
1766:
1689:
1663:
1035:
1644:
1133:
603:
1593:
937:
246:
1164:
1493:
966:
414:, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods
1790:
1499:
936:", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see
1260:
1118:
369:
1160:
Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.
1330:
1253:
773:
1412:
1171:
1122:
1071:
1021:
824:
779:
1056:
are perfectly normal
Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
998:
1476:
Counterexamples to some variations on these statements can be found in the lists above. Specifically,
1053:
981:
1650:
Kemoto, Nobuyuki (2004). "Higher
Separation Axioms". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.).
1477:
1447:
1137:
1087:
1064:
899:
891:
473:
1470:
1435:
1374:
1111:
1107:
1083:
1017:
962:
726:
681:
45:
1181:
769:
1580:
1762:
1685:
1659:
1640:
1632:
1396:
1103:
1075:
344:
292:
270:
52:
31:
1739:
1712:
1655:
1400:
1355:
1252:. In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also
1141:
1099:
1010:
308:
60:
40:
1459:
1408:
1382:
1039:
986:
974:
450:
296:
168:
96:
17:
1060:
629:
609:
585:
565:
545:
1755:
649:
1779:
1681:
1673:
1392:
1249:
1193:
1149:
1028:
1006:
944:
485:
396:
355:
150:
1744:
1727:
1717:
1700:
1388:
In fact, any space that satisfies any one of these three conditions must be normal.
898:. Every perfectly normal space is completely normal, because perfect normality is a
1404:
1542:
1163:
The main significance of normal spaces lies in the fact that they admit "enough"
1167:
1145:
1046:
875:
262:
1366:
928:" and derived concepts occasionally have a different meaning. (Nonetheless, "T
358:
284:
1126:
1395:
of normal spaces is not necessarily normal. This fact was first proved by
1049:
Hausdorff spaces are normal, and all paracompact regular spaces are normal;
402:
1451:
1428:
887:
435:
258:
78:
1543:"Why are these two definitions of a perfectly normal space equivalent?"
1528:
Engelking, Theorem 1.5.19. This is stated under the assumption of a T
1442:" to "normal completely regular" is the same as what we usually call
1174:, as expressed by the following theorems valid for any normal space
387:
that are also disjoint. More intuitively, this condition says that
1024:) are perfectly normal regular, although not in general Hausdorff;
894:. The equivalence between these three characterizations is called
1001:
are normal
Hausdorff spaces, or at least normal regular spaces:
768:. This is a stronger separation property than normality, as by
1502: – Property of topological spaces stronger than normality
1496: – Property of topological spaces stronger than normality
1098:
An important example of a non-normal topology is given by the
1480:
is normal but not regular, while the space of functions from
422:, here represented by larger, but still disjoint, open disks.
866:, but not precisely separated in general. It turns out that
886:
is perfectly normal if and only if every closed set is the
1532:
space, but the proof does not make use of that assumption.
492:
is completely normal if and only if every open subset of
1117:
A non-normal space of some relevance to analysis is the
1333:
1090:
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
222:
202:
184:
167:
149:
131:
113:
95:
77:
59:
51:
39:
1754:
1701:"On the topological product of paracompact spaces"
1381:. This shows the relationship of normal spaces to
1345:
858:
813:
760:
715:
670:
638:
618:
594:
574:
554:
955:", or "completely normal Hausdorff" instead of "T
1419:axiom are preserved under arbitrary products.
1403:. 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
973:are discussed elsewhere; they are related to
8:
488:can be separated by neighbourhoods. Also,
1283:, then there exists a continuous function
1200:, then there exists a continuous function
1743:
1716:
1332:
924:Note that the terms "normal space" and "T
921:, is a perfectly normal Hausdorff space.
838:
826:
793:
781:
734:
728:
689:
683:
651:
631:
611:
587:
567:
547:
1423:Relationships to other separation axioms
1074: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
1616:
1576:
1564:
1512:
1484:to itself is Tychonoff but not normal.
1399:. An example of this phenomenon is the
1346:{\displaystyle \emptyset \rightarrow X}
1067:are hereditarily normal and Hausdorff.
496:is normal with the subspace topology.
445:that is normal; this is equivalent to
36:
1244:. In fact, we can take the values of
1045:Generalizing the above examples, all
480:is a normal space. It turns out that
7:
1728:"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
1639:, Heldermann Verlag Berlin, 1989.
1469:A topological space is said to be
1334:
1086:are normal (even if not regular).
313:completely normal Hausdorff spaces
25:
1462:. These are what we usually call
1134:topology of pointwise convergence
1013:) are perfectly normal Hausdorff;
604:precisely separated by a function
325:perfectly normal Hausdorff spaces
1786:Properties of topological spaces
1652:Encyclopedia of General Topology
938:History of the separation axioms
1761:. Reading, MA: Addison-Wesley.
1745:10.1090/S0002-9904-1948-09118-2
1718:10.1090/S0002-9904-1947-08858-3
1555:Engelking, Theorem 2.1.6, p. 68
1438:. Thus, anything from "normal R
1152:metric spaces is never normal.
1136:. More generally, a theorem of
1337:
1275:is a continuous function from
853:
847:
808:
802:
749:
743:
704:
698:
665:
653:
1:
1094:Examples of non-normal spaces
30:For normal vector space, see
1031:Hausdorff spaces are normal;
1594:"separation axioms in nLab"
1494:Collectionwise normal space
1036:Stone–Čech compactification
997:Most spaces encountered in
761:{\displaystyle f^{-1}(1)=F}
716:{\displaystyle f^{-1}(0)=E}
397:separated by neighbourhoods
1807:
1500:Monotonically normal space
1248:to be entirely within the
515:, is a completely normal T
29:
1753:Willard, Stephen (1970).
1699:Sorgenfrey, R.H. (1947).
1450:, we see that all normal
1377:precisely subordinate to
993:Examples of normal spaces
468:, is a topological space
464:hereditarily normal space
242:
27:Type of topological space
1261:Tietze extension theorem
1119:topological vector space
261:and related branches of
18:Normal topological space
1254:separated by a function
1022:pseudometrisable spaces
774:separated by a function
542:is a topological space
458:completely normal space
1347:
1267:is a closed subset of
1072:second-countable space
860:
815:
762:
717:
672:
640:
620:
596:
576:
556:
540:perfectly normal space
423:
145:(completely Hausdorff)
1732:Bull. Amer. Math. Soc
1726:Stone, A. H. (1948).
1705:Bull. Amer. Math. Soc
1519:Willard, Exercise 15C
1434:, then it is in fact
1427:If a normal space is
1348:
1054:topological manifolds
999: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
1448:Kolmogorov quotients
1365:is a locally finite
1331:
1259:More generally, the
1132:to itself, with the
1065:totally ordered sets
1042:is normal Hausdorff;
982:locally normal space
825:
780:
727:
682:
650:
630:
610:
586:
566:
546:
1413:Tychonoff's theorem
1138:Arthur Harold Stone
1110:, which is used in
1084:fully normal spaces
1034:In particular, the
1018:pseudometric spaces
963:Fully normal spaces
943:Terms like "normal
900:hereditary property
892:continuous function
163:(regular Hausdorff)
1633:Engelking, Ryszard
1436:completely regular
1375:partition of unity
1373:, then there is a
1369:of a normal space
1343:
1299:in the sense that
1196:closed subsets of
1112:algebraic geometry
1108: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
1791:Separation axioms
1768:978-0-486-43479-7
1691:978-0-13-181629-9
1674:Munkres, James R.
1665:978-0-444-50355-8
1397:Robert Sorgenfrey
1208:to the real line
1104:algebraic variety
1011: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
1798:
1772:
1760:
1757:General Topology
1749:
1747:
1722:
1720:
1695:
1680:(2nd ed.).
1669:
1656:Elsevier Science
1637:General Topology
1620:
1614:
1608:
1607:
1605:
1604:
1590:
1584:
1574:
1568:
1562:
1556:
1553:
1547:
1546:
1539:
1533:
1526:
1520:
1517:
1478:Sierpiński space
1464:normal Hausdorff
1401:Sorgenfrey plane
1356:lifting property
1352:
1350:
1349:
1344:
1146:uncountably many
1140:states that the
1100:Zariski topology
1088:Sierpiński space
1061:order topologies
1052: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:
21:
1806:
1805:
1801:
1800:
1799:
1797:
1796:
1795:
1776:
1775:
1769:
1752:
1738:(10): 977–982.
1725:
1698:
1692:
1672:
1666:
1649:
1629:
1624:
1623:
1615:
1611:
1602:
1600:
1592:
1591:
1587:
1575:
1571:
1563:
1559:
1554:
1550:
1541:
1540:
1536:
1531:
1527:
1523:
1518:
1514:
1509:
1490:
1455:
1441:
1432:
1425:
1418:
1409:Tychonoff plank
1383:paracompactness
1329:
1328:
1182:Urysohn's lemma
1158:
1096:
1040:Tychonoff space
1020:(and hence all
1009:(and hence all
995:
987:Nemytskii plane
975:paracompactness
970:
958:
954:
950:
935:
931:
927:
918:
910:
879:
834:
823:
822:
789:
778:
777:
770:Urysohn's lemma
730:
725:
724:
685:
680:
679:
648:
647:
628:
627:
608:
607:
584:
583:
564:
563:
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:
1804:
1802:
1794:
1793:
1788:
1778:
1777:
1774:
1773:
1767:
1750:
1723:
1711:(6): 631–632.
1696:
1690:
1670:
1664:
1647:
1628:
1625:
1622:
1621:
1609:
1585:
1569:
1557:
1548:
1534:
1529:
1521:
1511:
1510:
1508:
1505:
1504:
1503:
1497:
1489:
1486:
1453:
1444:normal regular
1439:
1430:
1424:
1421:
1416:
1342:
1339:
1336:
1236:) = 1 for all
1220:) = 0 for all
1157:
1154:
1095:
1092:
1080:
1079:
1076:Lindelöf space
1070:Every regular
1068:
1057:
1050:
1043:
1032:
1025:
1014:
994:
991:
968:
956:
952:
948:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1803:
1792:
1789:
1787:
1784:
1783:
1781:
1770:
1764:
1759:
1758:
1751:
1746:
1741:
1737:
1733:
1729:
1724:
1719:
1714:
1710:
1706:
1702:
1697:
1693:
1687:
1683:
1682:Prentice-Hall
1679:
1675:
1671:
1667:
1661:
1657:
1654:. Amsterdam:
1653:
1648:
1646:
1645:3-88538-006-4
1642:
1638:
1634:
1631:
1630:
1626:
1619:, Section 17.
1618:
1613:
1610:
1599:
1595:
1589:
1586:
1582:
1578:
1573:
1570:
1567:, p. 213
1566:
1561:
1558:
1552:
1549:
1544:
1538:
1535:
1525:
1522:
1516:
1513:
1506:
1501:
1498:
1495:
1492:
1491:
1487:
1485:
1483:
1479:
1474:
1472:
1467:
1465:
1461:
1457:
1449:
1445:
1437:
1433:
1422:
1420:
1414:
1410:
1406:
1402:
1398:
1394:
1389:
1386:
1384:
1380:
1376:
1372:
1368:
1364:
1359:
1357:
1353:
1340:
1324:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1295:that extends
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1257:
1255:
1251:
1250:unit interval
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1177:
1173:
1169:
1166:
1161:
1155:
1153:
1151:
1147:
1143:
1139:
1135:
1131:
1128:
1124:
1120:
1115:
1113:
1109:
1105:
1101:
1093:
1091:
1089:
1085:
1077:
1073:
1069:
1066:
1062:
1058:
1055:
1051:
1048:
1044:
1041:
1037:
1033:
1030:
1026:
1023:
1019:
1015:
1012:
1008:
1007:metric spaces
1004:
1003:
1002:
1000:
992:
990:
988:
983:
978:
976:
972:
964:
960:
946:
945:regular space
941:
939:
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:
338:
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:
1756:
1735:
1731:
1708:
1704:
1677:
1651:
1636:
1617:Willard 1970
1612:
1601:. Retrieved
1597:
1588:
1577:Willard 1970
1572:
1565:Munkres 2000
1560:
1551:
1537:
1524:
1515:
1481:
1475:
1471:pseudonormal
1468:
1463:
1443:
1426:
1390:
1387:
1378:
1370:
1362:
1360:
1327:
1325:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1258:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
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883:
871:
867:
539:
537:
528:
524:
520:
509:completely T
508:
500:
498:
493:
489:
481:
477:
469:
461:
457:
455:
446:
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425:
419:
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392:
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384:
380:
376:
372:
368:, there are
365:
361:
352:normal space
351:
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342:
328:
324:
316:
312:
300:
288:
277:
273:
267:normal space
266:
256:
223:
203:
185:
133:completely T
73:(Kolmogorov)
1598:ncatlab.org
1579:, pp.
1415:) and the T
1047: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)
1780:Categories
1627:References
1603:2021-10-12
1367:open cover
1315:) for all
1212:such that
1165:continuous
1156:Properties
1106:or on the
1082:Also, all
1078:is normal.
678:such that
53:Kolmogorov
1507:Citations
1460:Tychonoff
1446:. Taking
1338:→
1335:∅
1172:functions
1127:real line
1125:from the
1123:functions
840:−
832:⊆
795:−
787:⊆
736:−
691:−
451:Hausdorff
127:(Urysohn)
91:(Fréchet)
1678:Topology
1676:(2000).
1488:See also
1466:spaces.
1354:has the
1326:The map
1194:disjoint
1192:are two
1170:-valued
888:zero set
474:subspace
356:disjoint
259:topology
1581:100–101
1393:product
1150:compact
1142:product
1121:of all
1029:compact
967:fully T
602:can be
535:space.
395:can be
278:Axiom T
247:History
173:3½
1765:
1688:
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1456:spaces
1405:Dowker
1102:on an
971:spaces
519:space
333:spaces
323:, and
321:spaces
232:
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141:
120:½
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1263:: If
1204:from
1184:: If
1038:of a
919:space
913:, or
911:space
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626:from
513:space
507:, or
505:space
460:, or
440:space
434:is a
432:space
350:is a
327:, or
315:, or
305:space
269:is a
1763:ISBN
1686:ISBN
1660:ISBN
1641:ISBN
1458:are
1307:) =
1271:and
1228:and
1188:and
1168:real
1148:non-
1059:All
1027:All
1016:All
1005:All
965:and
821:and
723:and
582:and
418:and
410:and
391:and
379:and
364:and
265:, a
1740:doi
1713:doi
1361:If
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1279:to
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959:".
880:set
476:of
383:of
375:of
287:of
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1596:.
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1234:x
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969:4
957:5
953:4
949:4
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930:5
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909:6
907:T
884:X
878:δ
876:G
872:X
868:X
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744:(
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438:1
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420:V
416:U
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319:5
317:T
303:4
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289:X
280:4
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227:6
224:T
207:5
204:T
189:4
186:T
169:T
154:3
151:T
136:2
118:2
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100:2
97:T
82:1
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64:0
61:T
34:.
20:)
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