31:
267:
may be mapped injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual
Euclidean metric are the canonical example.
470:
1236:
570:
669:
501:
384:
82:
1105:
963:
1066:
616:
924:
843:
1027:
998:
877:
810:
723:
1278:
1128:
693:
392:
1163:
523:
621:
741:
17:
1137:
is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its
122:
30:
1239:
1435:
477:
336:
37:
134:
1071:
929:
1032:
583:
893:
815:
1003:
970:
849:
782:
696:
291:
702:
8:
1245:
1352:
1138:
1113:
737:
678:
295:
1408:
322:
318:
153:
107:
1110:
Informally, these conditions means that every digit of the binary representation of
1411:
92:
1130:
that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.
1378:
1142:
730:
273:
252:
192:
1347:
573:
232:
1429:
772:
1389:, Escuela Regional de Matemáticas. Universidad del Valle, Colombia: 145–147
302:
188:
1357:
465:{\displaystyle S=\{0\}\cup \{1,{\tfrac {1}{2}},{\tfrac {1}{3}},\dots \},}
283:
260:
173:
88:
251:, since the isolation of each of its points together with the fact that
1325:. It follows that each point of the Cantor set lies in the closure of
1157:
278:
277:(every neighbourhood of a point contains other points of the set). A
1416:
326:
256:
248:
220:
208:
145:
1379:"An explicit set of isolated points in R with uncountable closure"
1342:
27:
Point of a subset S around which there are no other points of S
1310:
is any point in the Cantor set, then every neighborhood of
509:
is not an isolated point because there are other points in
287:(it contains all its limit points and no isolated points).
1152:
with the same properties can be obtained as follows. Let
1029:
then exactly one of the following two conditions holds:
1406:
1231:{\displaystyle I_{1},I_{2},I_{3},\ldots ,I_{k},\ldots }
482:
439:
424:
1248:
1166:
1116:
1074:
1035:
1006:
973:
932:
896:
852:
818:
785:
705:
681:
624:
586:
526:
480:
395:
339:
40:
223:
that is made up only of isolated points is called a
775:representation fulfills the following conditions:
1272:
1230:
1122:
1099:
1060:
1021:
992:
957:
918:
871:
837:
804:
717:
687:
663:
610:
564:
495:
464:
378:
321:in the following three examples are considered as
305:, the number of isolated points in each is equal.
76:
747:
160:). Another equivalent formulation is: an element
1427:
565:{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}
1376:
725:(and is therefore, in some sense, "close" to
664:{\displaystyle \tau =\{\emptyset ,\{a\},X\},}
215:that contains only finitely many elements of
712:
706:
655:
646:
640:
631:
605:
593:
559:
535:
456:
414:
408:
402:
352:
346:
53:
47:
1284:be a set consisting of one point from each
271:A set with no isolated point is said to be
129:that does not contain any other points of
729:). Such a situation is not possible in a
528:
133:. This is equivalent to saying that the
29:
14:
1428:
1407:
1383:Matemáticas: Enseñanza universitaria
890:denotes the largest index such that
313:
744:of certain functions are isolated.
290:The number of isolated points is a
281:with no isolated point is called a
24:
1321:, and hence at least one point of
1306:is an isolated point. However, if
675:is an isolated point, even though
634:
238:
25:
1447:
1400:
386:the point 0 is an isolated point.
879:only for finitely many indices
496:{\displaystyle {\tfrac {1}{k}}}
1370:
1261:
1249:
748:Two counter-intuitive examples
742:non-degenerate critical points
370:
358:
71:
59:
13:
1:
1377:Gomez-Ramirez, Danny (2007),
1363:
1298:contains only one point from
379:{\displaystyle S=\{0\}\cup ,}
329:with the standard topology.
77:{\displaystyle A=\{0\}\cup }
34:"0" is an isolated point of
7:
1336:
308:
247:of Euclidean space must be
172:if and only if it is not a
10:
1452:
1100:{\displaystyle x_{i+1}=1.}
958:{\displaystyle x_{m-1}=0.}
505:is an isolated point, but
1333:has uncountable closure.
1061:{\displaystyle x_{i-1}=1}
611:{\displaystyle X=\{a,b\}}
580:In the topological space
263:means that the points of
148:in the topological space
919:{\displaystyle x_{m}=1,}
838:{\displaystyle x_{i}=1.}
203:is an isolated point of
168:is an isolated point of
1022:{\displaystyle i<m,}
993:{\displaystyle x_{i}=1}
872:{\displaystyle x_{i}=1}
805:{\displaystyle x_{i}=0}
1314:contains at least one
1274:
1232:
1124:
1101:
1062:
1023:
994:
959:
920:
873:
839:
806:
764:such that every digit
719:
689:
665:
612:
566:
497:
466:
380:
84:
78:
1275:
1233:
1156:be the middle-thirds
1125:
1102:
1063:
1024:
995:
960:
921:
874:
840:
807:
760:in the real interval
720:
718:{\displaystyle \{a\}}
690:
666:
613:
567:
498:
467:
381:
292:topological invariant
79:
33:
1246:
1164:
1114:
1072:
1033:
1004:
971:
930:
894:
850:
816:
783:
703:
679:
622:
584:
524:
478:
393:
337:
243:Any discrete subset
38:
472:each of the points
207:if there exists an
121:and there exists a
1409:Weisstein, Eric W.
1353:Accumulation point
1273:{\displaystyle -C}
1270:
1228:
1120:
1097:
1058:
1019:
990:
955:
916:
869:
835:
802:
715:
685:
661:
608:
576:is a discrete set.
562:
493:
491:
462:
448:
433:
376:
319:Topological spaces
296:topological spaces
229:discrete point set
195:, then an element
85:
74:
1302:, every point of
1123:{\displaystyle x}
752:Consider the set
688:{\displaystyle b}
490:
447:
432:
314:Standard examples
152:(considered as a
117:is an element of
108:topological space
16:(Redirected from
1443:
1436:General topology
1422:
1421:
1412:"Isolated Point"
1391:
1390:
1374:
1332:
1329:, and therefore
1328:
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1320:
1313:
1309:
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1020:
999:
997:
996:
991:
983:
982:
964:
962:
961:
956:
948:
947:
925:
923:
922:
917:
906:
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694:
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571:
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266:
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218:
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206:
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191:, for example a
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83:
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21:
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1197:
1193:
1184:
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1167:
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1162:
1161:
1153:
1149:
1143:uncountable set
1134:
1115:
1112:
1111:
1079:
1075:
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1069:
1040:
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1034:
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1005:
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1001:
978:
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928:
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857:
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847:
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813:
790:
786:
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781:
780:
769:
765:
761:
757:
753:
750:
731:Hausdorff space
726:
704:
701:
700:
695:belongs to the
680:
677:
676:
672:
623:
620:
619:
585:
582:
581:
574:natural numbers
527:
525:
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481:
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438:
423:
394:
391:
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338:
335:
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311:
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274:dense-in-itself
264:
244:
241:
239:Related notions
216:
212:
204:
200:
196:
193:Euclidean space
184:
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137:
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95:
39:
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28:
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15:
12:
11:
5:
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1402:
1401:External links
1399:
1393:
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1362:
1361:
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1350:
1348:Adherent point
1345:
1338:
1335:
1317:
1294:
1291:. Since each
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711:
708:
684:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
618:with topology
607:
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561:
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351:
348:
345:
342:
315:
312:
310:
307:
294:, i.e. if two
240:
237:
233:discrete space
100:isolated point
73:
70:
67:
64:
61:
58:
55:
52:
49:
46:
43:
26:
9:
6:
4:
3:
2:
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1404:
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1373:
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1349:
1346:
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1341:
1340:
1334:
1267:
1264:
1258:
1255:
1252:
1242:intervals of
1241:
1225:
1222:
1217:
1213:
1209:
1206:
1203:
1198:
1194:
1190:
1185:
1181:
1177:
1172:
1168:
1159:
1146:
1144:
1140:
1131:
1117:
1094:
1091:
1086:
1083:
1080:
1076:
1055:
1052:
1047:
1044:
1041:
1037:
1016:
1013:
1010:
1007:
987:
984:
979:
975:
966:
952:
949:
944:
941:
938:
934:
913:
910:
907:
902:
898:
885:
866:
863:
858:
854:
846:
832:
829:
824:
820:
799:
796:
791:
787:
778:
777:
776:
774:
745:
743:
739:
734:
732:
709:
698:
682:
658:
652:
649:
643:
637:
628:
625:
602:
599:
596:
590:
587:
575:
556:
553:
550:
547:
544:
541:
538:
532:
519:
487:
484:
459:
453:
450:
444:
441:
435:
429:
426:
420:
417:
411:
405:
399:
396:
388:
373:
367:
364:
361:
355:
349:
343:
340:
332:
331:
330:
328:
324:
320:
306:
304:
297:
293:
288:
286:
285:
280:
276:
275:
269:
262:
258:
254:
250:
236:
234:
230:
226:
222:
210:
194:
190:
183:If the space
181:
175:
155:
147:
141:
136:
124:
109:
101:
98:is called an
94:
90:
68:
65:
62:
56:
50:
44:
41:
32:
19:
1415:
1396:
1386:
1382:
1372:
1148:Another set
1147:
1132:
1109:
751:
740:states that
735:
671:the element
579:
513:as close to
389:For the set
333:For the set
317:
303:homeomorphic
289:
282:
272:
270:
242:
228:
225:discrete set
224:
189:metric space
182:
139:
123:neighborhood
102:of a subset
99:
86:
18:Discrete set
1358:Point cloud
738:Morse lemma
517:as desired.
284:perfect set
174:limit point
89:mathematics
1364:References
1280:, and let
1158:Cantor set
756:of points
279:closed set
231:(see also
1417:MathWorld
1265:−
1240:component
1226:…
1207:…
1045:−
942:−
771:of their
635:∅
626:τ
557:…
454:…
412:∪
356:∪
327:real line
323:subspaces
253:rationals
249:countable
221:point set
209:open ball
135:singleton
57:∪
1430:Category
1337:See also
520:The set
309:Examples
154:subspace
146:open set
1238:be the
1139:closure
779:Either
697:closure
503:
474:
325:of the
259:in the
211:around
1343:Acnode
1160:, let
1141:is an
773:binary
144:is an
106:(in a
1133:Now,
926:then
762:(0,1)
261:reals
257:dense
187:is a
113:) if
93:point
1011:<
1000:and
736:The
301:are
299:X, Y
255:are
219:. A
91:, a
1068:or
967:If
886:If
812:or
699:of
572:of
235:).
227:or
199:of
176:of
164:of
156:of
125:of
87:In
1432::
1414:.
1387:15
1385:,
1381:,
1145:.
1095:1.
953:0.
833:1.
733:.
180:.
142:}
1420:.
1331:F
1327:F
1323:F
1318:k
1316:I
1312:p
1308:p
1304:F
1300:F
1295:k
1293:I
1288:k
1286:I
1282:F
1268:C
1262:]
1259:1
1256:,
1253:0
1250:[
1223:,
1218:k
1214:I
1210:,
1204:,
1199:3
1195:I
1191:,
1186:2
1182:I
1178:,
1173:1
1169:I
1154:C
1150:F
1135:F
1118:x
1092:=
1087:1
1084:+
1081:i
1077:x
1056:1
1053:=
1048:1
1042:i
1038:x
1017:,
1014:m
1008:i
988:1
985:=
980:i
976:x
950:=
945:1
939:m
935:x
914:,
911:1
908:=
903:m
899:x
888:m
883:.
881:i
867:1
864:=
859:i
855:x
830:=
825:i
821:x
800:0
797:=
792:i
788:x
768:i
766:x
758:x
754:F
727:a
713:}
710:a
707:{
683:b
673:a
659:,
656:}
653:X
650:,
647:}
644:a
641:{
638:,
632:{
629:=
606:}
603:b
600:,
597:a
594:{
591:=
588:X
560:}
554:,
551:2
548:,
545:1
542:,
539:0
536:{
533:=
529:N
515:0
511:S
507:0
488:k
485:1
460:,
457:}
451:,
445:3
442:1
436:,
430:2
427:1
421:,
418:1
415:{
409:}
406:0
403:{
400:=
397:S
374:,
371:]
368:2
365:,
362:1
359:[
353:}
350:0
347:{
344:=
341:S
265:S
245:S
217:S
213:x
205:S
201:S
197:x
185:X
178:S
170:S
166:S
162:x
158:X
150:S
140:x
138:{
131:S
127:x
119:S
115:x
111:X
104:S
96:x
72:]
69:2
66:,
63:1
60:[
54:}
51:0
48:{
45:=
42:A
20:)
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