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1802:. Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points. Perfect sets are particularly important in applications of the
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of the space. In other words, the derived set is not changed by adding to or removing from the given set a finite number of points. It can also be shown that in a T
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Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in
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subset of a Polish space is the union of a countable set and a set that is perfect with respect to the
1441:(in the second space) of any subset of the first space is the image of the derived set of that subset.
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space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T
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This investigation into the derivation process was one of the motivations for introducing
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of a topological space is defined by repeatedly applying the derived set operation using
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2637:{\displaystyle \displaystyle X^{\lambda }=\bigcap _{\alpha <\lambda }X^{\alpha }}
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subset of a Polish space is again a Polish space, the theorem also shows that any G
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can be written as the union of a countable set and a perfect set. Because any
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But the derived set of a closed set is always closed. In addition, if
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2578:{\displaystyle \displaystyle X^{\alpha +1}=\left(X^{\alpha }\right)'}
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spaces, the derived set of any finite set is empty and furthermore,
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if every subset consisting of a single point is closed. In a T
2910:"General topology - Proving the derived set $ E'$ is closed"
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The transfinite sequence of Cantor–Bendixson derivatives of
3335:
PlanetMath's article on the Cantor–Bendixson derivative
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Pages displaying short descriptions of redirect targets
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and must eventually be constant. The smallest ordinal
2210:{\displaystyle (S\cup T)^{*}\subseteq S^{*}\cup T^{*}}
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665:then the derived set has the following properties:
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1370:and each is disjoint from the other's derived set
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1112:need not be closed in general. For example, if
572:{\displaystyle A'=\mathbb {R} \setminus \{1\}.}
984:contains all its limit points. For any subset
2346:will define a topology on the space in which
2009:{\displaystyle \varnothing ^{*}=\varnothing }
1543:{\displaystyle (S-\{p\})'=S'=(S\cup \{p\})',}
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2837: – Cluster point in a topological space
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1636:{\displaystyle \left(S'\right)'\subseteq S'}
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1420:{\textstyle S'\cap T=\varnothing =T'\cap S.}
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2144:{\displaystyle a\in (S\setminus \{a\})^{*}}
2059:{\displaystyle S^{**}\subseteq S^{*}\cup S}
116:in large part to study derived sets on the
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3103:take the derived set on both sides to get
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2935:Hocking, John G.; Young, Gail S. (1988) ,
692:{\displaystyle \varnothing '=\varnothing }
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2756:{\displaystyle X^{\alpha +1}=X^{\alpha }}
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2822: – a stronger analog of limit point
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773:{\displaystyle a\in (A\setminus \{a\})'}
1092:The derived set of a subset of a space
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2379:is the derived set operator, that is,
1437:if and only if the derived set of the
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2516:{\displaystyle \displaystyle X^{0}=X}
2280:{\displaystyle S^{*}\subseteq T^{*}.}
1277:, the derived set of every subset of
626:are subsets of the topological space
19:In mathematics, more specifically in
1433:between two topological spaces is a
833:{\displaystyle (A\cup B)'=A'\cup B'}
499:that contains 1. The derived set of
108:The concept was first introduced by
658:{\displaystyle (X,{\mathcal {F}}),}
13:
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14:
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1864:can be equipped with an operator
1836:Topology in terms of derived sets
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3287:Classical Descriptive Set Theory
3135:{\displaystyle S''\subseteq S';}
2339:{\displaystyle S^{*}\subseteq S}
16:Set of all limit points of a set
3266:Foundations of General Topology
3250:, vol. 1, Academic Press,
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1082:{\displaystyle {\overline {S}}}
3096:{\displaystyle S'\subseteq S,}
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2372:{\displaystyle S\mapsto S^{*}}
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1890:{\displaystyle S\mapsto S^{*}}
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957:{\displaystyle S'\subseteq S,}
896:{\displaystyle A'\subseteq B'}
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473:(open sets) consisting of the
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1:
3293:Graduate Texts in Mathematics
3228:. Heldermann Verlag, Berlin.
3196:
1712:{\displaystyle S\subseteq S'}
581:
123:
2237:{\displaystyle S\subseteq T}
1460:space). It follows that in T
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860:{\displaystyle A\subseteq B}
492:{\displaystyle \mathbb {R} }
462:{\displaystyle \mathbb {R} }
380:then the derived set of the
369:{\displaystyle \mathbb {R} }
7:
3264:Pervin, William J. (1964),
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349:
10:
3366:
3207:, Wm C. Brown Publishers,
2093:{\displaystyle a\in S^{*}}
926:of a topological space is
376:is endowed with its usual
3295:156 ed.). Springer.
2663:{\displaystyle \lambda .}
2413:{\displaystyle S^{*}=S'.}
1219:{\displaystyle S'=\{b\},}
1143:{\displaystyle X=\{a,b\}}
209:is the set of all points
112:in 1872 and he developed
78:It is usually denoted by
3205:Introduction to Topology
3203:Baker, Crump W. (1991),
2844:
2446:{\displaystyle \alpha ,}
1811:Cantor–Bendixson theorem
1366:precisely when they are
1031:{\displaystyle S\cup S'}
524:{\displaystyle A:=\{1\}}
2710:{\displaystyle \alpha }
2466:{\displaystyle \alpha }
1226:which is not closed in
1179:{\displaystyle S=\{a\}}
724:{\displaystyle a\in A'}
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3310:Sierpiński, Wacław F.
3281:Kechris, Alexander S.
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1897:mapping subsets of
202:{\displaystyle S',}
99:{\displaystyle S'.}
3222:Engelking, Ryszard
3183:{\displaystyle X.}
3180:
3160:{\displaystyle S'}
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3062:{\displaystyle X,}
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2787:{\displaystyle X.}
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1933:{\displaystyle X,}
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1844:. A set of points
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1242:{\displaystyle X.}
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1062:(that is, the set
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477:and any subset of
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378:Euclidean topology
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255:{\displaystyle S,}
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174:{\displaystyle X,}
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71:{\displaystyle S.}
68:
51:is the set of all
37:
21:point-set topology
3302:978-0-387-94374-9
3069:which shows that
3039:{\displaystyle S}
2941:, Dover, p.
2686:{\displaystyle X}
2607:
2303:{\displaystyle S}
1973:{\displaystyle a}
1953:{\displaystyle S}
1910:{\displaystyle X}
1857:{\displaystyle X}
1760:{\displaystyle S}
1732:{\displaystyle S}
1681:{\displaystyle S}
1583:{\displaystyle p}
1563:{\displaystyle S}
1355:{\displaystyle T}
1335:{\displaystyle S}
1290:{\displaystyle X}
1262:{\displaystyle X}
1105:{\displaystyle X}
1077:
1055:{\displaystyle S}
977:{\displaystyle S}
919:{\displaystyle S}
619:{\displaystyle B}
599:{\displaystyle A}
438:{\displaystyle .}
339:{\displaystyle x}
319:{\displaystyle S}
299:{\displaystyle x}
275:{\displaystyle x}
157:topological space
148:{\displaystyle S}
49:topological space
40:{\displaystyle S}
3357:
3350:General topology
3318:General Topology
3312:; translated by
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3268:, Academic Press
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863:
858:
839:
837:
836:
831:
829:
818:
807:
779:
777:
776:
771:
769:
730:
728:
727:
722:
720:
698:
696:
695:
690:
682:
664:
662:
661:
656:
648:
647:
625:
623:
622:
617:
605:
603:
602:
597:
578:
576:
575:
570:
553:
545:
530:
528:
527:
522:
498:
496:
495:
490:
488:
468:
466:
465:
460:
458:
444:
442:
441:
436:
412:
411:
406:
375:
373:
372:
367:
365:
345:
343:
342:
337:
325:
323:
322:
317:
305:
303:
302:
297:
282:such that every
281:
279:
278:
273:
262:that is, points
261:
259:
258:
253:
234:
232:
231:
226:
208:
206:
205:
200:
195:
180:
178:
177:
172:
154:
152:
151:
146:
105:
103:
102:
97:
92:
77:
75:
74:
69:
46:
44:
43:
38:
3365:
3364:
3360:
3359:
3358:
3356:
3355:
3354:
3340:
3339:
3331:
3303:
3276:
3274:Further reading
3258:
3236:
3215:
3199:
3194:
3193:
3172:
3169:
3168:
3149:
3147:
3144:
3143:
3121:
3110:
3108:
3105:
3104:
3076:
3074:
3071:
3070:
3051:
3048:
3047:
3031:
3028:
3027:
3022:
3018:
3008:
3007:
2999:
2995:
2989:Kuratowski 1966
2987:
2983:
2977:Kuratowski 1966
2975:
2971:
2963:
2959:
2953:
2933:
2929:
2921:
2917:
2908:
2907:
2903:
2895:
2891:
2883:
2879:
2871:
2867:
2859:
2852:
2847:
2838:
2823:
2810:
2798:ordinal numbers
2776:
2773:
2772:
2766:
2765:
2747:
2743:
2728:
2724:
2722:
2719:
2718:
2702:
2699:
2698:
2678:
2675:
2674:
2652:
2649:
2648:
2627:
2623:
2611:
2598:
2594:
2591:
2588:
2587:
2560:
2556:
2552:
2551:
2536:
2532:
2529:
2526:
2525:
2500:
2496:
2493:
2490:
2489:
2458:
2455:
2454:
2435:
2432:
2431:
2429:ordinal numbers
2425:
2399:
2390:
2386:
2384:
2381:
2380:
2363:
2359:
2351:
2348:
2347:
2324:
2320:
2318:
2315:
2314:
2295:
2292:
2291:
2268:
2264:
2255:
2251:
2249:
2246:
2245:
2223:
2220:
2219:
2201:
2197:
2188:
2184:
2175:
2171:
2157:
2154:
2153:
2135:
2131:
2105:
2102:
2101:
2084:
2080:
2072:
2069:
2068:
2044:
2040:
2028:
2024:
2022:
2019:
2018:
1994:
1990:
1988:
1985:
1984:
1965:
1962:
1961:
1945:
1942:
1941:
1922:
1919:
1918:
1902:
1899:
1898:
1881:
1877:
1869:
1866:
1865:
1849:
1846:
1845:
1838:
1827:
1822:
1780:
1772:
1769:
1768:
1752:
1749:
1748:
1745:dense-in-itself
1741:isolated points
1724:
1721:
1720:
1701:
1693:
1690:
1689:
1673:
1670:
1669:
1648:
1645:
1644:
1643:for any subset
1625:
1606:
1602:
1601:
1599:
1596:
1595:
1593:
1575:
1572:
1571:
1555:
1552:
1551:
1550:for any subset
1529:
1500:
1489:
1469:
1466:
1465:
1463:
1459:
1455:
1449:
1400:
1377:
1375:
1372:
1371:
1347:
1344:
1343:
1327:
1324:
1323:
1302:
1299:
1298:
1282:
1279:
1278:
1274:
1254:
1251:
1250:
1231:
1228:
1227:
1193:
1191:
1188:
1187:
1159:
1156:
1155:
1117:
1114:
1113:
1097:
1094:
1093:
1069:
1067:
1064:
1063:
1047:
1044:
1043:
1020:
1012:
1009:
1008:
989:
986:
985:
969:
966:
965:
937:
935:
932:
931:
930:precisely when
911:
908:
907:
885:
874:
872:
869:
868:
846:
843:
842:
822:
811:
800:
786:
783:
782:
762:
736:
733:
732:
713:
705:
702:
701:
675:
673:
670:
669:
643:
642:
631:
628:
627:
611:
608:
607:
591:
588:
587:
584:
549:
538:
536:
533:
532:
504:
501:
500:
484:
482:
479:
478:
454:
452:
449:
448:
415:
388:
385:
384:
361:
359:
356:
355:
352:
331:
328:
327:
311:
308:
307:
291:
288:
287:
267:
264:
263:
244:
241:
240:
214:
211:
210:
188:
186:
183:
182:
163:
160:
159:
140:
137:
136:
126:
85:
83:
80:
79:
60:
57:
56:
32:
29:
28:
17:
12:
11:
5:
3363:
3353:
3352:
3338:
3337:
3330:
3329:External links
3327:
3326:
3325:
3307:
3301:
3275:
3272:
3271:
3270:
3261:
3256:
3244:Kuratowski, K.
3240:
3234:
3218:
3213:
3198:
3195:
3192:
3191:
3179:
3176:
3155:
3152:
3131:
3127:
3124:
3120:
3116:
3113:
3092:
3089:
3086:
3082:
3079:
3058:
3055:
3035:
3015:
3014:
3006:
3005:
2993:
2981:
2969:
2957:
2951:
2927:
2915:
2901:
2897:Engelking 1989
2889:
2877:
2865:
2849:
2848:
2846:
2843:
2842:
2841:
2832:
2829:Isolated point
2826:
2817:
2814:Adherent point
2809:
2806:
2783:
2780:
2763:is called the
2750:
2746:
2742:
2737:
2734:
2731:
2727:
2706:
2682:
2671:
2670:
2659:
2656:
2646:limit ordinals
2630:
2626:
2620:
2617:
2614:
2610:
2606:
2601:
2597:
2585:
2572:
2568:
2563:
2559:
2555:
2550:
2545:
2542:
2539:
2535:
2523:
2511:
2508:
2503:
2499:
2462:
2442:
2439:
2424:
2421:
2409:
2405:
2402:
2398:
2393:
2389:
2366:
2362:
2358:
2355:
2335:
2332:
2327:
2323:
2312:
2299:
2290:Calling a set
2288:
2287:
2276:
2271:
2267:
2263:
2258:
2254:
2233:
2230:
2227:
2217:
2204:
2200:
2196:
2191:
2187:
2183:
2178:
2174:
2170:
2167:
2164:
2161:
2151:
2138:
2134:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2087:
2083:
2079:
2076:
2066:
2055:
2052:
2047:
2043:
2039:
2034:
2031:
2027:
2016:
2005:
2002:
1997:
1993:
1969:
1960:and any point
1949:
1929:
1926:
1917:to subsets of
1906:
1884:
1880:
1876:
1873:
1853:
1837:
1834:
1825:
1820:
1786:
1783:
1779:
1776:
1756:
1728:
1707:
1704:
1700:
1697:
1677:
1655:
1652:
1631:
1628:
1624:
1620:
1616:
1612:
1609:
1605:
1591:
1579:
1570:and any point
1559:
1539:
1535:
1532:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1506:
1503:
1499:
1495:
1492:
1488:
1485:
1482:
1479:
1476:
1473:
1461:
1457:
1453:
1447:
1416:
1413:
1410:
1406:
1403:
1399:
1396:
1393:
1390:
1387:
1383:
1380:
1351:
1331:
1309:
1306:
1286:
1272:
1258:
1238:
1235:
1215:
1212:
1209:
1206:
1203:
1199:
1196:
1175:
1172:
1169:
1166:
1163:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1101:
1076:
1073:
1051:
1026:
1023:
1019:
1016:
996:
993:
973:
964:that is, when
953:
950:
947:
943:
940:
915:
904:
903:
891:
888:
884:
880:
877:
856:
853:
850:
840:
828:
825:
821:
817:
814:
810:
806:
803:
799:
796:
793:
790:
780:
768:
765:
761:
758:
755:
752:
749:
746:
743:
740:
719:
716:
712:
709:
699:
688:
685:
681:
678:
654:
651:
646:
641:
638:
635:
615:
595:
583:
580:
568:
565:
562:
559:
556:
552:
548:
544:
541:
520:
517:
514:
511:
508:
487:
457:
434:
431:
428:
425:
422:
419:
404:
401:
398:
395:
392:
364:
351:
348:
335:
315:
295:
271:
251:
248:
224:
221:
218:
198:
194:
191:
170:
167:
144:
125:
122:
95:
91:
88:
67:
64:
36:
15:
9:
6:
4:
3:
2:
3362:
3351:
3348:
3347:
3345:
3336:
3333:
3332:
3323:
3319:
3315:
3311:
3308:
3304:
3298:
3294:
3289:
3288:
3282:
3278:
3277:
3267:
3262:
3259:
3257:0-12-429201-1
3253:
3249:
3245:
3241:
3237:
3235:3-88538-006-4
3231:
3227:
3223:
3219:
3216:
3214:0-697-05972-3
3210:
3206:
3201:
3200:
3177:
3174:
3167:is closed in
3153:
3150:
3129:
3125:
3122:
3118:
3114:
3111:
3090:
3087:
3084:
3080:
3077:
3056:
3053:
3033:
3025:
3020:
3016:
3013:
3012:
3002:
2997:
2990:
2985:
2978:
2973:
2966:
2961:
2954:
2952:0-486-65676-4
2948:
2944:
2940:
2939:
2931:
2924:
2919:
2911:
2905:
2898:
2893:
2886:
2881:
2874:
2869:
2862:
2857:
2855:
2850:
2836:
2833:
2830:
2827:
2821:
2818:
2815:
2812:
2811:
2805:
2803:
2799:
2794:
2781:
2778:
2770:
2748:
2744:
2740:
2735:
2732:
2729:
2725:
2704:
2696:
2680:
2657:
2654:
2647:
2628:
2624:
2618:
2615:
2612:
2608:
2604:
2599:
2595:
2586:
2570:
2566:
2561:
2557:
2553:
2548:
2543:
2540:
2537:
2533:
2524:
2509:
2506:
2501:
2497:
2488:
2487:
2486:
2484:
2480:
2478:
2460:
2440:
2437:
2430:
2420:
2407:
2403:
2400:
2396:
2391:
2387:
2364:
2360:
2353:
2333:
2330:
2325:
2321:
2310:
2297:
2274:
2269:
2265:
2261:
2256:
2252:
2231:
2228:
2225:
2218:
2202:
2198:
2194:
2189:
2185:
2181:
2176:
2168:
2165:
2162:
2152:
2136:
2125:
2116:
2110:
2107:
2085:
2081:
2077:
2074:
2067:
2053:
2050:
2045:
2041:
2037:
2032:
2029:
2025:
2017:
2000:
1995:
1991:
1983:
1982:
1981:
1967:
1947:
1927:
1924:
1904:
1882:
1878:
1871:
1851:
1843:
1833:
1831:
1823:
1816:
1812:
1807:
1805:
1801:
1784:
1781:
1777:
1774:
1754:
1746:
1742:
1726:
1705:
1702:
1698:
1695:
1675:
1666:
1653:
1650:
1629:
1626:
1622:
1618:
1614:
1610:
1607:
1603:
1577:
1557:
1537:
1533:
1523:
1517:
1514:
1508:
1504:
1501:
1497:
1493:
1483:
1477:
1474:
1451:
1444:A space is a
1442:
1440:
1436:
1435:homeomorphism
1432:
1427:
1414:
1411:
1408:
1404:
1401:
1397:
1391:
1388:
1385:
1381:
1378:
1369:
1365:
1349:
1329:
1320:
1307:
1304:
1297:is closed in
1284:
1276:
1256:
1236:
1233:
1213:
1207:
1201:
1197:
1194:
1170:
1164:
1161:
1153:
1134:
1131:
1128:
1122:
1119:
1099:
1090:
1071:
1049:
1041:
1024:
1021:
1017:
1014:
994:
991:
971:
951:
948:
945:
941:
938:
929:
913:
889:
886:
882:
878:
875:
854:
851:
848:
841:
826:
823:
819:
815:
812:
808:
804:
797:
794:
791:
781:
766:
756:
747:
741:
738:
717:
714:
710:
707:
700:
683:
679:
676:
668:
667:
666:
652:
639:
636:
613:
593:
579:
566:
560:
546:
542:
539:
515:
509:
506:
476:
472:
445:
432:
426:
423:
420:
399:
396:
393:
383:
379:
347:
333:
313:
293:
285:
284:neighbourhood
269:
249:
246:
238:
222:
219:
216:
196:
192:
189:
168:
165:
158:
142:
135:
131:
121:
119:
115:
111:
106:
93:
89:
86:
65:
62:
54:
50:
34:
26:
22:
3317:
3286:
3265:
3247:
3225:
3204:
3023:
3019:
3010:
3009:
2996:
2984:
2972:
2960:
2937:
2930:
2918:
2904:
2892:
2880:
2868:
2802:Georg Cantor
2795:
2764:
2672:
2485:as follows:
2474:
2426:
2289:
1839:
1815:Polish space
1808:
1798:is called a
1743:) is called
1739:contains no
1667:
1443:
1428:
1322:Two subsets
1321:
1091:
905:
585:
446:
353:
237:limit points
129:
127:
110:Georg Cantor
107:
53:limit points
27:of a subset
24:
18:
3001:Pervin 1964
2965:Pervin 1964
2923:Pervin 1964
2873:Pervin 1964
2835:Limit point
1800:perfect set
326:other than
181:denoted by
130:derived set
25:derived set
3197:References
2885:Baker 1991
2861:Baker 1991
2717:such that
2695:decreasing
2479:derivative
1719:(that is,
1154:, the set
582:Properties
124:Definition
114:set theory
3142:that is,
3119:⊆
3085:⊆
3026:Assuming
2749:α
2730:α
2705:α
2655:λ
2629:α
2619:λ
2613:α
2609:⋂
2600:λ
2562:α
2538:α
2477:Bendixson
2461:α
2438:α
2392:∗
2365:∗
2357:↦
2331:⊆
2326:∗
2270:∗
2262:⊆
2257:∗
2229:⊆
2203:∗
2195:∪
2190:∗
2182:⊆
2177:∗
2166:∪
2137:∗
2120:∖
2111:∈
2086:∗
2078:∈
2051:∪
2046:∗
2038:⊆
2033:∗
2030:∗
2004:∅
1996:∗
1992:∅
1883:∗
1875:↦
1699:⊆
1623:⊆
1518:∪
1478:−
1431:bijection
1409:∩
1395:∅
1386:∩
1364:separated
1150:with the
1075:¯
1018:∪
946:⊆
906:A subset
883:⊆
852:⊆
820:∪
795:∪
751:∖
742:∈
711:∈
687:∅
677:∅
555:∖
475:empty set
469:with the
447:Consider
235:that are
220:∈
118:real line
3344:Category
3316:(1952).
3283:(1995).
3248:Topology
3246:(1966),
3224:(1989).
3154:′
3126:′
3115:″
3081:′
2938:Topology
2808:See also
2571:′
2404:′
2244:implies
2100:implies
1842:topology
1785:′
1747:. A set
1706:′
1630:′
1619:′
1611:′
1534:′
1505:′
1494:′
1405:′
1382:′
1368:disjoint
1198:′
1025:′
1007:the set
942:′
890:′
879:′
867:implies
827:′
816:′
805:′
767:′
731:implies
718:′
680:′
543:′
471:topology
350:Examples
346:itself.
193:′
90:′
3003:, p. 62
2967:, p. 70
2925:, p. 51
2899:, p. 47
2887:, p. 42
2863:, p. 41
2475:Cantor–
1594:space,
1040:closure
3324:Press.
3299:
3254:
3232:
3211:
3024:Proof:
3011:Proofs
2991:, p.76
2979:, p.77
2949:
2875:, p.38
2311:closed
1668:A set
928:closed
134:subset
23:, the
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