488:
is open; hence any set, being the union of single points, is open. Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen.
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90:(and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "
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is open, which leaves the possibility of an open set whose complement is also open, making both sets both open
82:, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for
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have distance 1 if they're not the same point, and 0 otherwise. Under the resulting
970:
923:
Boolean algebra can be obtained in this way from a suitable topological space: see
539:
414:
This is a quite typical example: whenever a space is made up of a finite number of
737:
515:
445:
418:
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54:. That this is possible may seem counter-intuitive, as the common meanings of
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Any clopen set is a union of (possibly infinitely many) connected components.
485:
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35:
91:
51:
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47:
27:
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as operations, the clopen subsets of a given topological space
79:
740:
if and only if the only clopen sets are the empty set and
74:
closed, and therefore clopen. As described by topologist
62:
are antonyms, but their mathematical definitions are not
977:(2nd ed.). John Wiley & Sons, Inc. p. 348.
1085:
1072:{\displaystyle \operatorname {Bdry} (A)=\varnothing }
1047:
1027:
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if and only if it is a union of connected components.
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979:(regarding the real numbers and the empty set in R)
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Stone's representation theorem for
Boolean algebras
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1041:be a subset of a topological space. Prove that
969:
492:As a less trivial example, consider the space
888:if and only if all of its subsets are clopen.
988:
421:in this way, the components will be clopen.
940: – Equalities for combinations of sets
993:. NY: Dover Publications, Inc. p. 56.
518:with their ordinary topology, and the set
1004:
989:Hocking, John G.; Young, Gail S. (1961).
817:has only finitely many components, or if
690:
668:
619:
574:
500:
307:
251:
1115:
957:
538:of all positive rational numbers whose
1169:
542:is bigger than 2. Using the fact that
448: – that is, two points
1009:(Third ed.). Dover. p. 87.
938:List of set identities and relations
16:Subset which is both open and closed
763:A set is clopen if and only if its
682:; it is neither open nor closed in
13:
1153:
296:from the ordinary topology on the
86:is unrelated to their meaning for
14:
1188:
1066:
660:a clopen subset of the real line
973:; Sherbert, Donald R. (1992) .
591:one can show quite easily that
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1054:
998:
995:(regarding topological spaces)
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963:
398:
386:
366:
354:
230:
218:
198:
186:
1:
1109:
975:Introduction to Real Analysis
710:
700:{\displaystyle \mathbb {R} .}
629:{\displaystyle \mathbb {Q} .}
584:{\displaystyle \mathbb {Q} ,}
444:be an infinite set under the
317:{\displaystyle \mathbb {R} .}
261:{\displaystyle \mathbb {R} .}
675:{\displaystyle \mathbb {R} }
507:{\displaystyle \mathbb {Q} }
7:
931:
841:), then a set is clopen in
797:are open (for instance, if
559:{\displaystyle {\sqrt {2}}}
97:
10:
1193:
66:. A set is closed if its
18:
1005:Mendelson, Bert (1990) .
379:is clopen, as is the set
102:In any topological space
1156:"Topology Without Tears"
1007:Introduction to Topology
944:
473:{\displaystyle p,q\in X}
19:Not to be confused with
152:Now consider the space
46:is a set which is both
1125:Upper Saddle River, NJ
1093:
1073:
1035:
909:
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855:
831:
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791:
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611:is a clopen subset of
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438:
408:
407:{\displaystyle (2,3).}
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172:which consists of the
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1102:(Given as Exercise 7)
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372:{\displaystyle (0,1)}
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236:{\displaystyle (2,3)}
206:
204:{\displaystyle (0,1)}
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120:
1083:
1045:
1025:
899:
891:Using the union and
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864:A topological space
845:
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775:connected components
744:
720:
716:A topological space
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419:connected components
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292:is inherited as the
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129:and the whole space
106:
1123:(Second ed.).
1099:is open and closed.
1154:Morris, Sidney A.
1129:Prentice Hall, Inc
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340:{\displaystyle X,}
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139:
118:{\displaystyle X,}
115:
64:mutually exclusive
21:Half-open interval
1162:on 19 April 2013.
1138:978-0-13-181629-9
1117:Munkres, James R.
1092:{\displaystyle A}
1034:{\displaystyle A}
971:Bartle, Robert G.
908:{\displaystyle X}
877:{\displaystyle X}
854:{\displaystyle X}
839:locally connected
830:{\displaystyle X}
810:{\displaystyle X}
790:{\displaystyle X}
753:{\displaystyle X}
729:{\displaystyle X}
649:{\displaystyle A}
604:{\displaystyle A}
554:
531:{\displaystyle A}
437:{\displaystyle X}
294:subspace topology
285:{\displaystyle X}
165:{\displaystyle X}
149:are both clopen.
142:{\displaystyle X}
44:topological space
1184:
1177:General topology
1163:
1158:. Archived from
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176:of the two open
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1079:if and only if
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917:Boolean algebra
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446:discrete metric
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94:" their name.
40:closed-open set
24:
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9:
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1016:0-486-66352-3
1012:
1008:
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992:
985:
976:
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960:, p. 91.
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708:
694:
657:
643:
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551:
541:
525:
517:
490:
487:
486:singleton set
483:
467:
464:
461:
458:
455:
447:
431:
422:
420:
417:
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389:
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95:
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87:
83:
81:
77:
76:James Munkres
71:
69:
65:
59:
55:
53:
49:
45:
41:
37:
33:
29:
22:
1160:the original
1120:
1020:
1006:
1000:
990:
984:
974:
965:
958:Munkres 2000
953:
893:intersection
491:
482:metric space
423:
151:
101:
39:
31:
25:
92:door spaces
78:, unlike a
36:portmanteau
1110:References
711:Properties
566:is not in
68:complement
32:clopen set
1067:∅
1052:
767:is empty.
738:connected
465:∈
298:real line
178:intervals
127:empty set
1171:Category
1147:42683260
1121:Topology
1119:(2000).
991:Topology
932:See also
886:discrete
765:boundary
424:Now let
416:disjoint
347:the set
270:topology
98:Examples
28:topology
915:form a
773:If all
760:itself.
514:of all
42:) in a
1145:
1135:
1013:
540:square
484:, any
60:closed
52:closed
945:Notes
921:Every
174:union
84:doors
1143:OCLC
1133:ISBN
1049:Bdry
1021:Let
1011:ISBN
268:The
211:and
125:the
88:sets
80:door
58:and
56:open
50:and
48:open
30:, a
884:is
837:is
777:of
736:is
658:not
656:is
324:In
272:on
243:of
72:and
38:of
34:(a
26:In
1173::
1141:.
1131:.
1127::
1019:.
919:.
707:)
1149:.
1087:A
1064:=
1061:)
1058:A
1055:(
1029:A
927:.
903:X
872:X
849:X
825:X
805:X
785:X
748:X
724:X
695:.
691:R
669:R
644:A
636:(
624:.
620:Q
599:A
579:,
575:Q
552:2
526:A
501:Q
468:X
462:q
459:,
456:p
432:X
402:.
399:)
396:3
393:,
390:2
387:(
367:)
364:1
361:,
358:0
355:(
335:,
332:X
312:.
308:R
280:X
256:.
252:R
231:)
228:3
225:,
222:2
219:(
199:)
196:1
193:,
190:0
187:(
160:X
137:X
113:,
110:X
23:.
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