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53:. 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
1049:
894:
870:
738:
488:
460:
291:
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199:
171:
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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
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1409:
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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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1427:
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205:
76:
43:
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668:
Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every
815:
1054:
1335:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
873:
209:
643:
438:
Every totally separated space is evidently totally disconnected but the converse is false even for
1367:
1300:
1028:
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463:
989:
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Confusingly, in the literature (for instance) totally disconnected spaces are sometimes called
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1371:
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1314:
1304:
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39:
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It is in general not true that every open set in a totally disconnected space is also closed.
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1381:
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is a totally disconnected
Hausdorff space that does not have small inductive dimension 0.
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108:
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553:
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263:
65:
693:
Every totally disconnected compact metric space is homeomorphic to a subset of a
624:
31:
676:
549:
61:
497:
the two notions (totally disconnected and totally separated) are equivalent.
1354:
1318:
694:
658:
27:
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is totally disconnected but its quasicomponents are not singletons. For
716:
Constructing a totally disconnected quotient space of any given space
267:
50:
60:
An important example of a totally disconnected space is the
1267:
1265:
516:
The following are examples of totally disconnected spaces:
876:
whose equivalence classes are the connected components of
1023:
totally disconnected quotient but in a certain sense the
153:
are the one-point sets. Analogously, a topological space
986:
continuous. With a little bit of effort we can see that
654:
of totally disconnected spaces are totally disconnected.
1248:. Heldermann Verlag, Sigma Series in Pure Mathematics.
1148:
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111:
1135:{\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}
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1043:
1008:
978:
918:
888:
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804:
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454:
427:
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351:, there is a pair of disjoint open neighborhoods
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311:
285:
254:
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165:
145:
117:
319:. Equivalently, for each pair of distinct points
852:denotes the largest connected subset containing
16:Topological space that is maximally disconnected
1404:(Revised ed.), New York: Academic Press ,
608:{\displaystyle \,\cap \,\mathbb {Q} ^{\omega }}
212:are singletons. That is, a topological space
979:{\displaystyle m:x\mapsto \mathrm {conn} (x)}
8:
690:0 if and only if it is totally disconnected.
306:
300:
204:Another closely related notion is that of a
1271:
1031:holds: For any totally disconnected space
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1147:
1115:
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110:
75:. Another example, playing a key role in
1228:
1209:
805:{\displaystyle y\in \mathrm {conn} (x)}
740:be an arbitrary topological space. Let
508:is used for totally separated spaces.
1216:
1176:{\displaystyle f={\breve {f}}\circ m}
7:
1337:McGraw-Hill Science/Engineering/Math
541:-adic numbers; more generally, all
963:
960:
957:
954:
845:{\displaystyle \mathrm {conn} (x)}
829:
826:
823:
820:
789:
786:
783:
780:
14:
1433:Properties of topological spaces
1402:Topology II: Transl. from French
1076:{\displaystyle f:X\rightarrow Y}
657:Totally disconnected spaces are
1389:(reprint of the 1970 original,
1019:In fact this space is not only
684:locally compact Hausdorff space
1400:Kuratowski, Kazimierz (1968),
1126:
1123:
1109:
1067:
973:
967:
950:
839:
833:
799:
793:
665:, since singletons are closed.
1:
1193:Extremally disconnected space
675:is a continuous image of the
21:extremally disconnected space
470:with the apex removed. Then
428:{\displaystyle X=U\sqcup V}
1449:
1219:, p. 395 Appendix A7.
1198:Totally disconnected group
1016:is totally disconnected.
36:totally disconnected space
18:
1362:Willard, Stephen (2004),
1009:{\displaystyle X/{\sim }}
919:{\displaystyle X/{\sim }}
688:small inductive dimension
574:0 is totally disconnected
572:small inductive dimension
570:Every Hausdorff space of
545:are totally disconnected.
504:, while the terminology
502:hereditarily disconnected
175:totally path-disconnected
872:). This is obviously an
344:{\displaystyle x,y\in X}
201:are the one-point sets.
30:and related branches of
19:Not to be confused with
1051:and any continuous map
759:{\displaystyle x\sim y}
709:extremally disconnected
619:Extremally disconnected
206:totally separated space
77:algebraic number theory
1297:Upper Saddle River, NJ
1177:
1136:
1077:
1045:
1010:
980:
920:
890:
866:
846:
806:
760:
734:
631:Knaster–Kuratowski fan
609:
484:
468:Knaster–Kuratowski fan
456:
429:
397:
371:
345:
313:
287:
256:
255:{\displaystyle x\in X}
226:
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147:
119:
1178:
1137:
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1011:
981:
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891:
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457:
442:. For instance, take
430:
398:
372:
346:
314:
312:{\displaystyle \{x\}}
288:
257:
227:
208:, i.e. a space where
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168:
148:
120:
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1091:
1055:
1035:
990:
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874:equivalence relation
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816:
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506:totally disconnected
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381:
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323:
297:
277:
240:
216:
185:
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137:
131:connected components
127:totally disconnected
109:
105:A topological space
1332:Functional Analysis
1295:(Second ed.).
396:{\displaystyle x,y}
370:{\displaystyle U,V}
57:connected subsets.
1368:Dover Publications
1301:Prentice Hall, Inc
1242:Engelking, Ryszard
1173:
1132:
1073:
1041:
1029:universal property
1006:
976:
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533:irrational numbers
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452:
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191:
163:
143:
115:
1377:978-0-486-43479-7
1346:978-0-07-054236-5
1310:978-0-13-181629-9
1289:Munkres, James R.
1164:
1103:
1083:, there exists a
1044:{\displaystyle Y}
928:quotient topology
889:{\displaystyle X}
865:{\displaystyle x}
733:{\displaystyle X}
483:{\displaystyle X}
455:{\displaystyle X}
293:is the singleton
286:{\displaystyle x}
234:totally separated
225:{\displaystyle X}
194:{\displaystyle X}
166:{\displaystyle X}
146:{\displaystyle X}
118:{\displaystyle X}
40:topological space
1440:
1428:General topology
1414:
1388:
1364:General topology
1358:
1322:
1275:
1269:
1260:
1259:
1246:General Topology
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1096:
1087:continuous map
1082:
1080:
1079:
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1027:: The following
1015:
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849:
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739:
737:
736:
731:
621:Hausdorff spaces
614:
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543:profinite groups
527:rational numbers
495:Hausdorff spaces
489:
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1412:
1399:
1378:
1361:
1347:
1325:
1311:
1287:
1284:
1279:
1278:
1274:, pp. 151.
1272:Kuratowski 1968
1270:
1263:
1256:
1240:
1239:
1235:
1231:, pp. 152.
1227:
1223:
1215:
1211:
1206:
1189:
1144:
1143:
1089:
1088:
1053:
1052:
1033:
1032:
988:
987:
936:
935:
934:making the map
932:finest topology
898:
897:
878:
877:
854:
853:
814:
813:
768:
767:
766:if and only if
742:
741:
722:
721:
718:
699:discrete spaces
662:
640:
593:
583:
582:
566:Sorgenfrey line
521:Discrete spaces
514:
492:locally compact
472:
471:
466:, which is the
464:Cantor's teepee
444:
443:
405:
404:
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295:
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275:
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214:
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210:quasicomponents
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179:path-components
155:
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135:
134:
107:
106:
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88:
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79:, is the field
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12:
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73:-adic integers
68:to the set of
42:that has only
15:
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9:
6:
4:
3:
2:
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1411:9780124292024
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1327:Rudin, Walter
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1255:3-88538-006-4
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498:
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477:
469:
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449:
441:
440:metric spaces
436:
422:
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410:
390:
387:
384:
364:
361:
358:
338:
335:
332:
329:
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303:
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272:
271:neighborhoods
269:
265:
249:
246:
243:
236:if for every
235:
219:
211:
207:
202:
188:
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176:
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140:
132:
128:
112:
100:
98:
96:
95:-adic numbers
94:
87:
83:
78:
74:
72:
67:
63:
58:
56:
52:
49:
45:
41:
37:
33:
29:
22:
1401:
1363:
1331:
1292:
1245:
1236:
1229:Munkres 2000
1224:
1212:
1084:
1024:
1020:
1018:
719:
673:metric space
625:Stone spaces
554:Cantor space
538:
515:
505:
501:
499:
437:
264:intersection
233:
203:
174:
126:
104:
92:
85:
81:
70:
66:homeomorphic
59:
54:
35:
25:
930:, i.e. the
697:product of
579:Erdős space
560:Baire space
64:, which is
32:mathematics
1422:Categories
1282:References
1217:Rudin 1991
677:Cantor set
652:coproducts
638:Properties
550:Cantor set
462:to be the
403:such that
101:Definition
62:Cantor set
44:singletons
1204:Citations
1168:∘
1162:˘
1127:→
1121:∼
1101:˘
1068:→
1003:∼
951:↦
926:with the
913:∼
777:∈
751:∼
695:countable
644:Subspaces
601:ω
590:∩
420:⊔
336:∈
247:∈
48:connected
1355:21163277
1329:(1991).
1319:42683260
1293:Topology
1291:(2000).
1244:(1989).
1187:See also
896:. Endow
648:products
552:and the
512:Examples
177:if all
28:topology
1394:0264581
1386:2048350
1025:biggest
812:(where
670:compact
266:of all
129:if the
51:subsets
1408:
1384:
1374:
1353:
1343:
1317:
1307:
1252:
1085:unique
663:spaces
650:, and
268:clopen
262:, the
1142:with
38:is a
1406:ISBN
1372:ISBN
1351:OCLC
1341:ISBN
1315:OCLC
1305:ISBN
1250:ISBN
1021:some
720:Let
686:has
629:The
577:The
564:The
558:The
548:The
537:The
531:The
525:The
55:only
34:, a
1183:.
377:of
273:of
232:is
181:in
173:is
133:in
125:is
90:of
46:as
26:In
1424::
1391:MR
1382:MR
1380:,
1370:,
1366:,
1349:.
1339:.
1313:.
1303:.
1299::
1264:^
682:A
646:,
435:.
97:.
1396:)
1357:.
1321:.
1258:.
1171:m
1159:f
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1150:f
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1124:)
1117:/
1113:X
1110:(
1107::
1098:f
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1065:X
1062::
1059:f
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998:/
994:X
974:)
971:x
968:(
964:n
961:n
958:o
955:c
948:x
945::
942:m
908:/
904:X
884:X
860:x
840:)
837:x
834:(
830:n
827:n
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800:)
797:x
794:(
790:n
787:n
784:o
781:c
774:y
754:y
748:x
728:X
711:.
701:.
679:.
661:1
659:T
596:Q
581:â„“
539:p
478:X
450:X
423:V
417:U
414:=
411:X
391:y
388:,
385:x
365:V
362:,
359:U
339:X
333:y
330:,
327:x
307:}
304:x
301:{
281:x
250:X
244:x
220:X
189:X
161:X
141:X
113:X
93:p
86:p
82:Q
71:p
23:.
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