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64:. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the
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It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected
Hausdorff space is
220:
1443:
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1319:
1347:
17:
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provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.
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31:
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Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every
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1346:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
698:
662:
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449:
Every totally separated space is evidently totally disconnected but the converse is false even for
417:
216:
87:
54:
1307:
1000:
910:
511:
Confusingly, in the literature (for instance) totally disconnected spaces are sometimes called
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It is in general not true that every open set in a totally disconnected space is also closed.
80:
250:
1404:
1396:
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307:
8:
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is a totally disconnected
Hausdorff space that does not have small inductive dimension 0.
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50:
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1401:
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Every totally disconnected compact metric space is homeomorphic to a subset of a
576:
553:
537:
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141:
58:
709:
531:
589:
1432:
1299:
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the two notions (totally disconnected and totally separated) are equivalent.
102:
1365:
1329:
1337:
683:
564:
450:
274:
76:
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42:
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is totally disconnected but its quasicomponents are not singletons. For
687:
560:
72:
727:
Constructing a totally disconnected quotient space of any given space
705:
669:
38:
1221:
278:
61:
71:
An important example of a totally disconnected space is the
1278:
1276:
527:
The following are examples of totally disconnected spaces:
887:
whose equivalence classes are the connected components of
1034:
totally disconnected quotient but in a certain sense the
164:
are the one-point sets. Analogously, a topological space
1273:
997:
continuous. With a little bit of effort we can see that
665:
of totally disconnected spaces are totally disconnected.
1259:. Heldermann Verlag, Sigma Series in Pure Mathematics.
1233:
1159:
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1146:{\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}
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1145:
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989:
929:
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875:
855:
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406:
380:
362:, there is a pair of disjoint open neighborhoods
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322:
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204:
176:
156:
128:
330:. Equivalently, for each pair of distinct points
1430:
863:denotes the largest connected subset containing
27:Topological space that is maximally disconnected
1415:(Revised ed.), New York: Academic Press ,
619:{\displaystyle \,\cap \,\mathbb {Q} ^{\omega }}
223:are singletons. That is, a topological space
990:{\displaystyle m:x\mapsto \mathrm {conn} (x)}
701:0 if and only if it is totally disconnected.
317:
311:
215:Another closely related notion is that of a
1410:
1282:
1042:holds: For any totally disconnected space
1251:
606:
603:
599:
86:. Another example, playing a key role in
1372:
1298:
1239:
816:{\displaystyle y\in \mathrm {conn} (x)}
751:be an arbitrary topological space. Let
14:
1431:
519:is used for totally separated spaces.
1336:
1227:
1187:{\displaystyle f={\breve {f}}\circ m}
1348:McGraw-Hill Science/Engineering/Math
552:-adic numbers; more generally, all
24:
974:
971:
968:
965:
856:{\displaystyle \mathrm {conn} (x)}
840:
837:
834:
831:
800:
797:
794:
791:
25:
1455:
1444:Properties of topological spaces
1413:Topology II: Transl. from French
1087:{\displaystyle f:X\rightarrow Y}
668:Totally disconnected spaces are
1400:(reprint of the 1970 original,
1030:In fact this space is not only
695:locally compact Hausdorff space
1411:Kuratowski, Kazimierz (1968),
1245:
1137:
1134:
1120:
1078:
984:
978:
961:
850:
844:
810:
804:
676:, since singletons are closed.
13:
1:
1292:
1204:Extremally disconnected space
686:is a continuous image of the
648:
111:
32:extremally disconnected space
1214:
481:with the apex removed. Then
7:
1197:
522:
439:{\displaystyle X=U\sqcup V}
10:
1460:
1230:, p. 395 Appendix A7.
1209:Totally disconnected group
1027:is totally disconnected.
47:totally disconnected space
29:
1373:Willard, Stephen (2004),
1020:{\displaystyle X/{\sim }}
930:{\displaystyle X/{\sim }}
699:small inductive dimension
585:0 is totally disconnected
583:small inductive dimension
581:Every Hausdorff space of
556:are totally disconnected.
515:, while the terminology
513:hereditarily disconnected
186:totally path-disconnected
883:). This is obviously an
355:{\displaystyle x,y\in X}
212:are the one-point sets.
41:and related branches of
30:Not to be confused with
1062:and any continuous map
770:{\displaystyle x\sim y}
720:extremally disconnected
630:Extremally disconnected
217:totally separated space
88:algebraic number theory
1308:Upper Saddle River, NJ
1188:
1147:
1088:
1056:
1021:
991:
931:
901:
877:
857:
817:
771:
745:
642:Knaster–Kuratowski fan
620:
495:
479:Knaster–Kuratowski fan
467:
440:
408:
382:
356:
324:
298:
267:
266:{\displaystyle x\in X}
237:
206:
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158:
130:
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1089:
1057:
1022:
992:
932:
902:
878:
858:
818:
772:
746:
621:
496:
468:
453:. For instance, take
441:
409:
383:
357:
325:
323:{\displaystyle \{x\}}
299:
268:
238:
219:, i.e. a space where
207:
179:
159:
131:
1157:
1102:
1066:
1046:
1001:
949:
911:
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885:equivalence relation
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827:
781:
755:
735:
596:
517:totally disconnected
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457:
418:
392:
366:
334:
308:
288:
251:
227:
196:
168:
148:
142:connected components
138:totally disconnected
120:
116:A topological space
18:Totally disconnected
1343:Functional Analysis
1306:(Second ed.).
407:{\displaystyle x,y}
381:{\displaystyle U,V}
68:connected subsets.
1379:Dover Publications
1312:Prentice Hall, Inc
1253:Engelking, Ryszard
1184:
1143:
1084:
1052:
1040:universal property
1017:
987:
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813:
767:
741:
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544:irrational numbers
491:
463:
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233:
202:
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126:
1388:978-0-486-43479-7
1357:978-0-07-054236-5
1321:978-0-13-181629-9
1300:Munkres, James R.
1175:
1114:
1094:, there exists a
1055:{\displaystyle Y}
939:quotient topology
900:{\displaystyle X}
876:{\displaystyle x}
744:{\displaystyle X}
494:{\displaystyle X}
466:{\displaystyle X}
304:is the singleton
297:{\displaystyle x}
245:totally separated
236:{\displaystyle X}
205:{\displaystyle X}
177:{\displaystyle X}
157:{\displaystyle X}
129:{\displaystyle X}
51:topological space
16:(Redirected from
1451:
1439:General topology
1425:
1399:
1375:General topology
1369:
1333:
1286:
1280:
1271:
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1257:General Topology
1249:
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1098:continuous map
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1059:
1058:
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1038:: The following
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776:
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748:
747:
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632:Hausdorff spaces
625:
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614:
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554:profinite groups
538:rational numbers
506:Hausdorff spaces
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1389:
1358:
1322:
1295:
1290:
1289:
1285:, pp. 151.
1283:Kuratowski 1968
1281:
1274:
1267:
1250:
1246:
1242:, pp. 152.
1238:
1234:
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1222:
1217:
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1002:
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964:
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947:
946:
945:making the map
943:finest topology
922:
917:
912:
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888:
868:
865:
864:
830:
828:
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790:
782:
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777:if and only if
756:
753:
752:
736:
733:
732:
729:
710:discrete spaces
673:
651:
610:
605:
604:
597:
594:
593:
577:Sorgenfrey line
532:Discrete spaces
525:
503:locally compact
486:
483:
482:
477:, which is the
475:Cantor's teepee
458:
455:
454:
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221:quasicomponents
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190:path-components
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121:
118:
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114:
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90:, is the field
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11:
5:
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84:-adic integers
79:to the set of
53:that has only
26:
9:
6:
4:
3:
2:
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1422:9780124292024
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1338:Rudin, Walter
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507:
504:
488:
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460:
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451:metric spaces
447:
433:
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424:
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401:
398:
395:
375:
372:
369:
349:
346:
343:
340:
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314:
291:
283:
282:neighborhoods
280:
276:
260:
257:
254:
247:if for every
246:
230:
222:
218:
213:
199:
191:
187:
171:
151:
143:
139:
123:
109:
107:
106:-adic numbers
105:
98:
94:
89:
85:
83:
78:
74:
69:
67:
63:
60:
56:
52:
48:
44:
40:
33:
19:
1412:
1374:
1342:
1303:
1256:
1247:
1240:Munkres 2000
1235:
1223:
1095:
1035:
1031:
1029:
730:
684:metric space
636:Stone spaces
565:Cantor space
549:
526:
516:
512:
510:
448:
275:intersection
244:
214:
185:
137:
115:
103:
96:
92:
81:
77:homeomorphic
70:
65:
46:
36:
941:, i.e. the
708:product of
590:Erdős space
571:Baire space
75:, which is
43:mathematics
1433:Categories
1293:References
1228:Rudin 1991
688:Cantor set
663:coproducts
649:Properties
561:Cantor set
473:to be the
414:such that
112:Definition
73:Cantor set
55:singletons
1215:Citations
1179:∘
1173:˘
1138:→
1132:∼
1112:˘
1079:→
1014:∼
962:↦
937:with the
924:∼
788:∈
762:∼
706:countable
655:Subspaces
612:ω
601:∩
431:⊔
347:∈
258:∈
59:connected
1366:21163277
1340:(1991).
1330:42683260
1304:Topology
1302:(2000).
1255:(1989).
1198:See also
907:. Endow
659:products
563:and the
523:Examples
188:if all
39:topology
1405:0264581
1397:2048350
1036:biggest
823:(where
681:compact
277:of all
140:if the
62:subsets
1419:
1395:
1385:
1364:
1354:
1328:
1318:
1263:
1096:unique
674:spaces
661:, and
279:clopen
273:, the
1153:with
49:is a
1417:ISBN
1383:ISBN
1362:OCLC
1352:ISBN
1326:OCLC
1316:ISBN
1261:ISBN
1032:some
731:Let
697:has
640:The
588:The
575:The
569:The
559:The
548:The
542:The
536:The
66:only
45:, a
1194:.
388:of
284:of
243:is
192:in
184:is
144:in
136:is
101:of
57:as
37:In
1435::
1402:MR
1393:MR
1391:,
1381:,
1377:,
1360:.
1350:.
1324:.
1314:.
1310::
1275:^
693:A
657:,
446:.
108:.
1407:)
1368:.
1332:.
1269:.
1182:m
1170:f
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1161:f
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1128:/
1124:X
1121:(
1118::
1109:f
1082:Y
1076:X
1073::
1070:f
1050:Y
1009:/
1005:X
985:)
982:x
979:(
975:n
972:n
969:o
966:c
959:x
956::
953:m
919:/
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848:x
845:(
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808:x
805:(
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798:n
795:o
792:c
785:y
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759:x
739:X
722:.
712:.
690:.
672:1
670:T
607:Q
592:â„“
550:p
489:X
461:X
434:V
428:U
425:=
422:X
402:y
399:,
396:x
376:V
373:,
370:U
350:X
344:y
341:,
338:x
318:}
315:x
312:{
292:x
261:X
255:x
231:X
200:X
172:X
152:X
124:X
104:p
97:p
93:Q
82:p
34:.
20:)
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