4212:
3995:
4233:
4201:
4270:
4243:
4223:
2646:. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. This is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than
2000:
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844:
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266:
112:
3108:
3060:
4273:
3759:
2444:
Hausdorff space into a compact
Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.
3868:
3831:
3801:
3771:
927:
2447:
Furthermore, every closed subset of a compact space is compact, and every compact subspace of a
Hausdorff space is closed.
671:, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space
421:
3907:
2437:
1795:
4261:
4256:
3744:
3712:
667:, instead of all nets. One value of this characterization is that it may be used as a definition in the context of
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810:
4251:
503:
1660:
1719:
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359:
4153:
216:
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3327:
is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense.
4161:
3197:
3113:
4294:
2670:
2637:
3960:
2866:
2673:
of any family of closed sets is closed (this includes intersections of infinitely many closed sets)
2441:
2221:
2055:
3351:
3288:
2772:
2005:
1451:
1378:
1312:
892:
784:
395:
306:
299:. Yet another equivalent definition is that a set is closed if and only if it contains all of its
158:
4246:
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2328:
978:
55:
28:
4181:
4102:
3979:
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3238:
3068:
3028:
2834:
184:
120:
4176:
3376:
1889:
4023:
3950:
1995:{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}(f(A))}
660:
75:
71:
24:
2175:
1125:
4171:
4123:
4097:
3945:
3582:
3513:
3243:
2794:
2621:
2279:
2123:
1232:
887:
2250:
8:
4018:
2643:
1919:
613:
300:
79:
4222:
3619:
3486: – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
3448:
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2419:
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2201:
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251:
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1008:
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549:
115:
63:
51:
4236:
2450:
Closed sets also give a useful characterization of compactness: a topological space
3984:
3930:
3339:
2342:
Whether a set is closed depends on the space in which it is embedded. However, the
4043:
4038:
3850:
3151:
2346:
83:
4226:
4133:
4065:
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2490:
with empty intersection admits a finite subcollection with empty intersection.
2350:
1883:
1118:
a subset is closed if and only if it contains every point that is close to it.
4288:
4143:
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3841:
3811:
3781:
3736:
3728:
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3006:
2343:
2332:
2029:
1114:
this terminology allows for a plain
English description of closed subsets:
4128:
4048:
3994:
3692:
2324:
3878:
4138:
3704:
3276:
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3011:
3002:
can be constructed as the intersection of all of these closed supersets.
2470:
is compact if and only if every collection of nonempty closed subsets of
2336:
2323:, as well as for other spaces that carry topological structures, such as
296:
67:
43:
4082:
3972:
3483:
3477:
3324:
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have the properties listed above, then there exists a unique topology
4107:
2691:
295:
Equivalently, a set is closed if and only if it contains all of its
4092:
4060:
4009:
3916:
3501:
3346:
3331:
2956:
2316:
664:
663:(such as a metric space), it is enough to consider only convergent
545:
59:
39:
35:
20:
3237:
Some sets are neither open nor closed, for instance the half-open
3072:
for an explanation of the bracket and parenthesis set notation.)
544:
An alternative characterization of closed sets is available via
1837:
for some (or equivalently, for every) topological super-space
66:, a closed set can be defined as a set which contains all its
2353:", in the sense that, if you embed a compact Hausdorff space
3370:
is an infinite and unbounded closed set in the real numbers.
3194:(inclusive) is closed in the space of rational numbers, but
3885:
3110:
is closed in the metric space of real numbers, and the set
3425:
is continuous if and only if preimages of closed sets in
2888:
The intersection property also allows one to define the
3518:
Pages displaying short descriptions of redirect targets
3497:
Pages displaying short descriptions of redirect targets
3488:
Pages displaying short descriptions of redirect targets
968:{\displaystyle x\in \operatorname {cl} _{A\cup \{x\}}A}
3330:
Singleton points (and thus finite sets) are closed in
2315:
The notion of closed set is defined above in terms of
694:
because whether or not a sequence or net converges in
3645:
3622:
3590:
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3411:
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2008:
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398:
362:
339:
309:
278:
254:
219:
187:
161:
123:
100:
3516: – Connected open subset of a topological space
3495: – Connected open subset of a topological space
453:{\displaystyle A\subseteq \operatorname {cl} _{X}A.}
3766:. New Jersey: World Scientific Publishing Company.
3696:
3663:
3631:
3608:
3573:
3553:
3460:
3437:
3417:
3397:
3362:
3312:
3275:Some sets are both open and closed and are called
3264:
3226:
3186:
3166:
3142:
3102:
3054:
2994:
2974:
2947:
2935:which is defined as the smallest closed subset of
2927:
2904:
2880:
2855:
2823:
2803:
2783:
2761:
2741:
2713:
2658:
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2529:
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2408:
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2300:
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2239:
2210:
2190:
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2144:
2112:
2092:
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2044:
2020:
1994:
1910:
1872:
1849:
1829:
1784:
1764:
1744:
1708:
1688:
1649:
1629:
1609:
1586:
1566:
1530:
1510:
1486:
1466:
1440:
1425:be closed in the "larger" surrounding super-space
1413:
1393:
1367:
1347:
1327:
1297:
1270:
1250:
1223:
1203:
1180:
1160:
1140:
1106:
1083:
1063:
1043:
1023:
999:
967:
916:
878:
858:
838:
799:
769:
749:
729:
706:
686:
651:
628:
604:
584:
564:
534:
492:
472:
452:
410:
384:
348:
321:
287:
260:
240:
205:
173:
141:
106:
1168:if and only if there exists some net (valued) in
4286:
2517:if there exist disjoint, nonempty, open subsets
2436:; the "surrounding space" does not matter here.
1830:{\displaystyle A=X\cap \operatorname {cl} _{Y}A}
82:operation. This should not be confused with a
3405:is a function between topological spaces then
2052:is continuous if and only if for every subset
3901:
3758:
3616:and not on the whole surrounding space (e.g.
3005:Sets that can be constructed as the union of
2742:{\displaystyle \mathbb {F} \neq \varnothing }
1882:Closed sets can also be used to characterize
839:{\displaystyle x\in \operatorname {cl} _{X}A}
3658:
3652:
3603:
3597:
3480: – Subset which is both open and closed
1375:), which is how it is possible for a subset
994:
988:
954:
948:
908:
902:
3504: – Basic subset of a topological space
1538:is always a (potentially proper) subset of
535:{\displaystyle A=\operatorname {cl} _{X}A.}
23:. For a set closed under an operation, see
19:This article is about the complement of an
4269:
4242:
3908:
3894:
1689:{\displaystyle A=\operatorname {cl} _{X}A}
3818:
3510: – Open set containing a given point
3356:
3220:
3136:
2871:
2777:
2729:
2310:
1745:{\displaystyle \operatorname {cl} _{Y}A.}
1696:), it is nevertheless still possible for
1567:{\displaystyle \operatorname {cl} _{Y}A,}
385:{\displaystyle \operatorname {cl} _{X}A;}
89:
3824:Handbook of Analysis and Its Foundations
3788:
3848:
3727:
4287:
3018:sets. These sets need not be closed.
2863:are exactly those sets that belong to
714:depends on what points are present in
241:{\displaystyle X\setminus A\in \tau .}
3889:
3691:
2172:is continuous at a fixed given point
1122:In terms of net convergence, a point
3764:Convergence Foundations Of Topology
3699:Principles of Mathematical Analysis
13:
3304:
3234:is not closed in the real numbers.
2416:will always be a closed subset of
268:if and only if it is equal to its
14:
4306:
3826:. San Diego, CA: Academic Press.
3227:{\displaystyle \cap \mathbb {Q} }
3143:{\displaystyle \cap \mathbb {Q} }
2736:
2319:, a concept that makes sense for
1319:
1071:is thus the set of all points in
223:
165:
74:, a closed set is a set which is
4268:
4241:
4231:
4221:
4210:
4200:
4199:
3993:
2831:such that the closed subsets of
2373:in an arbitrary Hausdorff space
1235:of some other topological space
3721:
3685:
3639:or any other space containing
3541:In particular, whether or not
3535:
3389:
3307:
3292:
3259:
3247:
3213:
3201:
3129:
3117:
3097:
3085:
3049:
3037:
2850:
2838:
2642:A closed set contains its own
2292:
2286:
2263:
2257:
2136:
2130:
2100:maps points that are close to
1989:
1986:
1980:
1974:
1902:
1657:(which happens if and only if
200:
188:
136:
124:
1:
3678:
3009:many closed sets are denoted
2982:Specifically, the closure of
2881:{\displaystyle \mathbb {F} .}
2631:
2240:{\displaystyle A\subseteq X,}
2074:{\displaystyle A\subseteq X,}
1574:which denotes the closure of
3915:
3363:{\displaystyle \mathbb {Z} }
3313:{\displaystyle [1,+\infty )}
2784:{\displaystyle \mathbb {F} }
2120:to points that are close to
2021:{\displaystyle A\subseteq X}
1467:{\displaystyle A\subseteq X}
1394:{\displaystyle A\subseteq X}
1355:(although not an element of
1328:{\displaystyle Y\setminus X}
1031:). Because the closure of
917:{\displaystyle A\cup \{x\},}
800:{\displaystyle A\subseteq X}
616:of every net of elements of
411:{\displaystyle A\subseteq X}
322:{\displaystyle A\subseteq X}
174:{\displaystyle X\setminus A}
16:Complement of an open subset
7:
3796:. Boston: Allyn and Bacon.
3762:; Mynard, Frédéric (2016).
3671:as a topological subspace).
3664:{\displaystyle A\cup \{x\}}
3609:{\displaystyle A\cup \{x\}}
3471:
3021:
2628:consisting of closed sets.
2438:Stone–Čech compactification
1498:topological super-space of
1000:{\displaystyle A\cup \{x\}}
329:is always contained in its
10:
4311:
4162:Banach fixed-point theorem
3849:Willard, Stephen (2004) .
2769:such that the elements of
2635:
2028:; this can be reworded in
866:belongs to the closure of
42:, and related branches of
18:
4195:
4152:
4116:
4002:
3991:
3923:
2856:{\displaystyle (X,\tau )}
2638:Kuratowski closure axioms
2440:, a process that turns a
1716:to be a proper subset of
206:{\displaystyle (X,\tau )}
142:{\displaystyle (X,\tau )}
3528:
3398:{\displaystyle f:X\to Y}
2701:In fact, if given a set
2697:The whole set is closed.
2329:differentiable manifolds
2198:if and only if whenever
1911:{\displaystyle f:X\to Y}
94:By definition, a subset
1280:topological super-space
572:of a topological space
29:Closed (disambiguation)
4217:Mathematics portal
4117:Metrics and properties
4103:Second-countable space
3665:
3633:
3610:
3575:
3555:
3462:
3439:
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3399:
3364:
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3266:
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3104:
3069:Interval (mathematics)
3056:
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2976:
2949:
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2882:
2857:
2825:
2805:
2785:
2763:
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2715:
2687:closed sets is closed.
2660:
2614:
2594:
2571:
2551:
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2507:
2484:
2464:
2430:
2410:
2390:
2367:
2311:More about closed sets
2302:
2270:
2241:
2212:
2192:
2191:{\displaystyle x\in X}
2166:
2146:
2114:
2094:
2075:
2046:
2022:
1996:
1912:
1874:
1851:
1831:
1786:
1772:is a closed subset of
1766:
1746:
1710:
1690:
1651:
1637:is a closed subset of
1631:
1611:
1588:
1568:
1532:
1512:
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1442:
1415:
1395:
1369:
1349:
1329:
1299:
1272:
1252:
1225:
1205:
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1162:
1142:
1141:{\displaystyle x\in X}
1108:
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969:
918:
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860:
840:
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771:
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731:
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536:
494:
480:is a closed subset of
474:
454:
412:
386:
350:
323:
289:
262:
242:
207:
175:
143:
108:
90:Equivalent definitions
27:. For other uses, see
3666:
3634:
3611:
3576:
3556:
3463:
3440:
3420:
3400:
3365:
3315:
3267:
3265:{\displaystyle [0,1)}
3229:
3189:
3169:
3145:
3105:
3057:
2997:
2977:
2950:
2930:
2907:
2883:
2858:
2826:
2806:
2804:{\displaystyle \tau }
2786:
2764:
2744:
2716:
2661:
2615:
2595:
2572:
2552:
2532:
2508:
2485:
2465:
2431:
2411:
2391:
2368:
2303:
2301:{\displaystyle f(A).}
2271:
2242:
2218:is close to a subset
2213:
2193:
2167:
2147:
2145:{\displaystyle f(A).}
2115:
2095:
2076:
2047:
2023:
1997:
1913:
1875:
1852:
1832:
1787:
1767:
1747:
1711:
1691:
1652:
1632:
1612:
1589:
1569:
1533:
1513:
1489:
1469:
1443:
1416:
1396:
1370:
1350:
1330:
1300:
1273:
1253:
1226:
1206:
1183:
1163:
1148:is close to a subset
1143:
1109:
1086:
1066:
1046:
1026:
1002:
970:
919:
881:
861:
846:(or equivalently, if
841:
802:
772:
752:
732:
709:
689:
661:first-countable space
654:
631:
612:if and only if every
607:
587:
567:
537:
495:
475:
455:
413:
387:
351:
331:(topological) closure
324:
290:
263:
243:
208:
181:is an open subset of
176:
144:
109:
72:complete metric space
25:closure (mathematics)
4172:Invariance of domain
4124:Euler characteristic
4098:Bundle (mathematics)
3643:
3620:
3588:
3581:depends only on the
3565:
3545:
3514:Region (mathematics)
3449:
3429:
3409:
3377:
3352:
3289:
3272:in the real numbers.
3244:
3198:
3178:
3158:
3114:
3082:
3034:
2986:
2963:
2939:
2916:
2896:
2867:
2835:
2815:
2795:
2773:
2753:
2725:
2705:
2650:
2622:totally disconnected
2604:
2581:
2561:
2541:
2521:
2497:
2493:A topological space
2474:
2454:
2420:
2400:
2377:
2357:
2280:
2269:{\displaystyle f(x)}
2251:
2222:
2202:
2176:
2156:
2124:
2104:
2084:
2056:
2036:
2006:
1926:
1890:
1884:continuous functions
1861:
1841:
1796:
1776:
1756:
1720:
1700:
1661:
1641:
1621:
1598:
1578:
1542:
1522:
1502:
1478:
1452:
1429:
1405:
1379:
1359:
1339:
1313:
1309:exist some point in
1286:
1262:
1239:
1233:topological subspace
1215:
1192:
1172:
1152:
1126:
1095:
1075:
1055:
1035:
1015:
1007:is endowed with the
979:
928:
893:
888:topological subspace
870:
850:
811:
785:
761:
741:
718:
698:
675:
640:
620:
596:
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556:
504:
484:
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422:
396:
360:
356:which is denoted by
337:
307:
276:
252:
217:
185:
159:
121:
98:
4182:Tychonoff's theorem
4177:Poincaré conjecture
3931:General (point-set)
248:A set is closed in
4167:De Rham cohomology
4088:Polyhedral complex
4078:Simplicial complex
3861:Dover Publications
3661:
3632:{\displaystyle X,}
3629:
3606:
3571:
3551:
3523:Regular closed set
3461:{\displaystyle X.}
3458:
3435:
3415:
3395:
3360:
3310:
3262:
3224:
3184:
3164:
3140:
3100:
3052:
2992:
2975:{\displaystyle A.}
2972:
2945:
2928:{\displaystyle X,}
2925:
2902:
2878:
2853:
2821:
2801:
2781:
2759:
2739:
2711:
2659:{\displaystyle 2.}
2656:
2610:
2593:{\displaystyle X.}
2590:
2567:
2547:
2527:
2503:
2480:
2460:
2442:completely regular
2426:
2406:
2389:{\displaystyle X,}
2386:
2363:
2321:topological spaces
2298:
2266:
2237:
2208:
2188:
2162:
2142:
2110:
2090:
2071:
2042:
2018:
1992:
1908:
1873:{\displaystyle X.}
1870:
1847:
1827:
1782:
1762:
1742:
1706:
1686:
1647:
1627:
1610:{\displaystyle Y;}
1607:
1584:
1564:
1528:
1508:
1484:
1464:
1441:{\displaystyle Y.}
1438:
1411:
1391:
1365:
1345:
1325:
1298:{\displaystyle X,}
1295:
1268:
1251:{\displaystyle Y,}
1248:
1221:
1204:{\displaystyle x.}
1201:
1188:that converges to
1178:
1158:
1138:
1107:{\displaystyle A,}
1104:
1091:that are close to
1081:
1061:
1041:
1021:
997:
965:
914:
876:
856:
836:
797:
767:
747:
730:{\displaystyle X.}
727:
704:
687:{\displaystyle X,}
684:
669:convergence spaces
652:{\displaystyle A.}
649:
626:
602:
582:
562:
532:
490:
470:
450:
408:
382:
349:{\displaystyle X,}
346:
319:
288:{\displaystyle X.}
285:
258:
238:
203:
171:
155:if its complement
139:
104:
4282:
4281:
4071:fundamental group
3870:978-0-486-43479-7
3833:978-0-12-622760-4
3803:978-0-697-06889-7
3773:978-981-4571-52-4
3729:Munkres, James R.
3574:{\displaystyle A}
3554:{\displaystyle x}
3438:{\displaystyle Y}
3418:{\displaystyle f}
3187:{\displaystyle 1}
3167:{\displaystyle 0}
2995:{\displaystyle X}
2948:{\displaystyle X}
2905:{\displaystyle A}
2824:{\displaystyle X}
2762:{\displaystyle X}
2721:and a collection
2714:{\displaystyle X}
2613:{\displaystyle X}
2570:{\displaystyle X}
2550:{\displaystyle B}
2530:{\displaystyle A}
2506:{\displaystyle X}
2483:{\displaystyle X}
2463:{\displaystyle X}
2429:{\displaystyle X}
2409:{\displaystyle D}
2366:{\displaystyle D}
2351:absolutely closed
2211:{\displaystyle x}
2165:{\displaystyle f}
2113:{\displaystyle A}
2093:{\displaystyle f}
2045:{\displaystyle f}
2002:for every subset
1850:{\displaystyle Y}
1785:{\displaystyle X}
1765:{\displaystyle A}
1709:{\displaystyle A}
1650:{\displaystyle X}
1630:{\displaystyle A}
1587:{\displaystyle A}
1531:{\displaystyle A}
1511:{\displaystyle X}
1487:{\displaystyle Y}
1414:{\displaystyle X}
1368:{\displaystyle X}
1348:{\displaystyle A}
1335:that is close to
1271:{\displaystyle Y}
1224:{\displaystyle X}
1181:{\displaystyle A}
1161:{\displaystyle A}
1084:{\displaystyle X}
1064:{\displaystyle X}
1044:{\displaystyle A}
1024:{\displaystyle X}
1011:induced on it by
1009:subspace topology
879:{\displaystyle A}
859:{\displaystyle x}
770:{\displaystyle X}
750:{\displaystyle x}
707:{\displaystyle X}
629:{\displaystyle A}
605:{\displaystyle X}
585:{\displaystyle X}
565:{\displaystyle A}
493:{\displaystyle X}
473:{\displaystyle A}
261:{\displaystyle X}
116:topological space
107:{\displaystyle A}
64:topological space
4302:
4295:General topology
4272:
4271:
4245:
4244:
4235:
4225:
4215:
4214:
4203:
4202:
3997:
3910:
3903:
3896:
3887:
3886:
3882:
3852:General Topology
3845:
3815:
3785:
3751:
3750:
3735:(2nd ed.).
3725:
3719:
3718:
3702:
3689:
3672:
3670:
3668:
3667:
3662:
3638:
3636:
3635:
3630:
3615:
3613:
3612:
3607:
3580:
3578:
3577:
3572:
3560:
3558:
3557:
3552:
3539:
3519:
3498:
3489:
3467:
3465:
3464:
3459:
3444:
3442:
3441:
3436:
3424:
3422:
3421:
3416:
3404:
3402:
3401:
3396:
3369:
3367:
3366:
3361:
3359:
3340:Hausdorff spaces
3319:
3317:
3316:
3311:
3271:
3269:
3268:
3263:
3233:
3231:
3230:
3225:
3223:
3193:
3191:
3190:
3185:
3173:
3171:
3170:
3165:
3152:rational numbers
3149:
3147:
3146:
3141:
3139:
3109:
3107:
3106:
3103:{\displaystyle }
3101:
3066:is closed. (See
3061:
3059:
3058:
3055:{\displaystyle }
3053:
3001:
2999:
2998:
2993:
2981:
2979:
2978:
2973:
2954:
2952:
2951:
2946:
2934:
2932:
2931:
2926:
2911:
2909:
2908:
2903:
2887:
2885:
2884:
2879:
2874:
2862:
2860:
2859:
2854:
2830:
2828:
2827:
2822:
2810:
2808:
2807:
2802:
2790:
2788:
2787:
2782:
2780:
2768:
2766:
2765:
2760:
2748:
2746:
2745:
2740:
2732:
2720:
2718:
2717:
2712:
2665:
2663:
2662:
2657:
2619:
2617:
2616:
2611:
2599:
2597:
2596:
2591:
2576:
2574:
2573:
2568:
2556:
2554:
2553:
2548:
2536:
2534:
2533:
2528:
2512:
2510:
2509:
2504:
2489:
2487:
2486:
2481:
2469:
2467:
2466:
2461:
2435:
2433:
2432:
2427:
2415:
2413:
2412:
2407:
2395:
2393:
2392:
2387:
2372:
2370:
2369:
2364:
2347:Hausdorff spaces
2307:
2305:
2304:
2299:
2275:
2273:
2272:
2267:
2246:
2244:
2243:
2238:
2217:
2215:
2214:
2209:
2197:
2195:
2194:
2189:
2171:
2169:
2168:
2163:
2151:
2149:
2148:
2143:
2119:
2117:
2116:
2111:
2099:
2097:
2096:
2091:
2080:
2078:
2077:
2072:
2051:
2049:
2048:
2043:
2027:
2025:
2024:
2019:
2001:
1999:
1998:
1993:
1970:
1969:
1957:
1953:
1946:
1945:
1917:
1915:
1914:
1909:
1879:
1877:
1876:
1871:
1856:
1854:
1853:
1848:
1836:
1834:
1833:
1828:
1820:
1819:
1791:
1789:
1788:
1783:
1771:
1769:
1768:
1763:
1751:
1749:
1748:
1743:
1732:
1731:
1715:
1713:
1712:
1707:
1695:
1693:
1692:
1687:
1679:
1678:
1656:
1654:
1653:
1648:
1636:
1634:
1633:
1628:
1617:indeed, even if
1616:
1614:
1613:
1608:
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1554:
1553:
1537:
1535:
1534:
1529:
1517:
1515:
1514:
1509:
1493:
1491:
1490:
1485:
1473:
1471:
1470:
1465:
1447:
1445:
1444:
1439:
1420:
1418:
1417:
1412:
1401:to be closed in
1400:
1398:
1397:
1392:
1374:
1372:
1371:
1366:
1354:
1352:
1351:
1346:
1334:
1332:
1331:
1326:
1304:
1302:
1301:
1296:
1277:
1275:
1274:
1269:
1257:
1255:
1254:
1249:
1230:
1228:
1227:
1222:
1210:
1208:
1207:
1202:
1187:
1185:
1184:
1179:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1113:
1111:
1110:
1105:
1090:
1088:
1087:
1082:
1070:
1068:
1067:
1062:
1050:
1048:
1047:
1042:
1030:
1028:
1027:
1022:
1006:
1004:
1003:
998:
974:
972:
971:
966:
958:
957:
923:
921:
920:
915:
885:
883:
882:
877:
865:
863:
862:
857:
845:
843:
842:
837:
829:
828:
806:
804:
803:
798:
776:
774:
773:
768:
756:
754:
753:
748:
736:
734:
733:
728:
713:
711:
710:
705:
693:
691:
690:
685:
658:
656:
655:
650:
636:also belongs to
635:
633:
632:
627:
611:
609:
608:
603:
591:
589:
588:
583:
571:
569:
568:
563:
541:
539:
538:
533:
522:
521:
499:
497:
496:
491:
479:
477:
476:
471:
459:
457:
456:
451:
440:
439:
417:
415:
414:
409:
391:
389:
388:
383:
372:
371:
355:
353:
352:
347:
328:
326:
325:
320:
303:. Every subset
294:
292:
291:
286:
267:
265:
264:
259:
247:
245:
244:
239:
212:
210:
209:
204:
180:
178:
177:
172:
148:
146:
145:
140:
113:
111:
110:
105:
4310:
4309:
4305:
4304:
4303:
4301:
4300:
4299:
4285:
4284:
4283:
4278:
4209:
4191:
4187:Urysohn's lemma
4148:
4112:
3998:
3989:
3961:low-dimensional
3919:
3914:
3871:
3834:
3820:Schechter, Eric
3804:
3790:Dugundji, James
3774:
3760:Dolecki, Szymon
3755:
3754:
3747:
3726:
3722:
3715:
3690:
3686:
3681:
3676:
3675:
3644:
3641:
3640:
3621:
3618:
3617:
3589:
3586:
3585:
3566:
3563:
3562:
3546:
3543:
3542:
3540:
3536:
3531:
3517:
3496:
3487:
3474:
3450:
3447:
3446:
3430:
3427:
3426:
3410:
3407:
3406:
3378:
3375:
3374:
3355:
3353:
3350:
3349:
3335:
3290:
3287:
3286:
3245:
3242:
3241:
3219:
3199:
3196:
3195:
3179:
3176:
3175:
3159:
3156:
3155:
3135:
3115:
3112:
3111:
3083:
3080:
3079:
3035:
3032:
3031:
3024:
3015:
2987:
2984:
2983:
2964:
2961:
2960:
2940:
2937:
2936:
2917:
2914:
2913:
2897:
2894:
2893:
2870:
2868:
2865:
2864:
2836:
2833:
2832:
2816:
2813:
2812:
2796:
2793:
2792:
2776:
2774:
2771:
2770:
2754:
2751:
2750:
2728:
2726:
2723:
2722:
2706:
2703:
2702:
2651:
2648:
2647:
2640:
2634:
2605:
2602:
2601:
2582:
2579:
2578:
2577:whose union is
2562:
2559:
2558:
2542:
2539:
2538:
2522:
2519:
2518:
2498:
2495:
2494:
2475:
2472:
2471:
2455:
2452:
2451:
2421:
2418:
2417:
2401:
2398:
2397:
2378:
2375:
2374:
2358:
2355:
2354:
2313:
2281:
2278:
2277:
2252:
2249:
2248:
2223:
2220:
2219:
2203:
2200:
2199:
2177:
2174:
2173:
2157:
2154:
2153:
2125:
2122:
2121:
2105:
2102:
2101:
2085:
2082:
2081:
2057:
2054:
2053:
2037:
2034:
2033:
2007:
2004:
2003:
1965:
1961:
1941:
1937:
1936:
1932:
1927:
1924:
1923:
1922:if and only if
1891:
1888:
1887:
1862:
1859:
1858:
1842:
1839:
1838:
1815:
1811:
1797:
1794:
1793:
1792:if and only if
1777:
1774:
1773:
1757:
1754:
1753:
1727:
1723:
1721:
1718:
1717:
1701:
1698:
1697:
1674:
1670:
1662:
1659:
1658:
1642:
1639:
1638:
1622:
1619:
1618:
1599:
1596:
1595:
1579:
1576:
1575:
1549:
1545:
1543:
1540:
1539:
1523:
1520:
1519:
1503:
1500:
1499:
1479:
1476:
1475:
1453:
1450:
1449:
1430:
1427:
1426:
1406:
1403:
1402:
1380:
1377:
1376:
1360:
1357:
1356:
1340:
1337:
1336:
1314:
1311:
1310:
1287:
1284:
1283:
1263:
1260:
1259:
1240:
1237:
1236:
1216:
1213:
1212:
1193:
1190:
1189:
1173:
1170:
1169:
1153:
1150:
1149:
1127:
1124:
1123:
1096:
1093:
1092:
1076:
1073:
1072:
1056:
1053:
1052:
1036:
1033:
1032:
1016:
1013:
1012:
980:
977:
976:
941:
937:
929:
926:
925:
894:
891:
890:
871:
868:
867:
851:
848:
847:
824:
820:
812:
809:
808:
786:
783:
782:
762:
759:
758:
742:
739:
738:
719:
716:
715:
699:
696:
695:
676:
673:
672:
641:
638:
637:
621:
618:
617:
597:
594:
593:
577:
574:
573:
557:
554:
553:
517:
513:
505:
502:
501:
500:if and only if
485:
482:
481:
465:
462:
461:
435:
431:
423:
420:
419:
397:
394:
393:
367:
363:
361:
358:
357:
338:
335:
334:
308:
305:
304:
301:boundary points
277:
274:
273:
253:
250:
249:
218:
215:
214:
186:
183:
182:
160:
157:
156:
122:
119:
118:
99:
96:
95:
92:
84:closed manifold
32:
17:
12:
11:
5:
4308:
4298:
4297:
4280:
4279:
4277:
4276:
4266:
4265:
4264:
4259:
4254:
4239:
4229:
4219:
4207:
4196:
4193:
4192:
4190:
4189:
4184:
4179:
4174:
4169:
4164:
4158:
4156:
4150:
4149:
4147:
4146:
4141:
4136:
4134:Winding number
4131:
4126:
4120:
4118:
4114:
4113:
4111:
4110:
4105:
4100:
4095:
4090:
4085:
4080:
4075:
4074:
4073:
4068:
4066:homotopy group
4058:
4057:
4056:
4051:
4046:
4041:
4036:
4026:
4021:
4016:
4006:
4004:
4000:
3999:
3992:
3990:
3988:
3987:
3982:
3977:
3976:
3975:
3965:
3964:
3963:
3953:
3948:
3943:
3938:
3933:
3927:
3925:
3921:
3920:
3913:
3912:
3905:
3898:
3890:
3884:
3883:
3869:
3846:
3832:
3816:
3802:
3786:
3772:
3753:
3752:
3745:
3720:
3713:
3683:
3682:
3680:
3677:
3674:
3673:
3660:
3657:
3654:
3651:
3648:
3628:
3625:
3605:
3602:
3599:
3596:
3593:
3570:
3550:
3533:
3532:
3530:
3527:
3526:
3525:
3520:
3511:
3505:
3499:
3490:
3481:
3473:
3470:
3469:
3468:
3457:
3454:
3445:are closed in
3434:
3414:
3394:
3391:
3388:
3385:
3382:
3371:
3358:
3343:
3333:
3328:
3321:
3309:
3306:
3303:
3300:
3297:
3294:
3280:
3273:
3261:
3258:
3255:
3252:
3249:
3235:
3222:
3218:
3215:
3212:
3209:
3206:
3203:
3183:
3163:
3138:
3134:
3131:
3128:
3125:
3122:
3119:
3099:
3096:
3093:
3090:
3087:
3073:
3071:
3051:
3048:
3045:
3042:
3039:
3023:
3020:
3013:
2991:
2971:
2968:
2944:
2924:
2921:
2901:
2877:
2873:
2852:
2849:
2846:
2843:
2840:
2820:
2800:
2779:
2758:
2749:of subsets of
2738:
2735:
2731:
2710:
2699:
2698:
2695:
2688:
2686:
2674:
2655:
2633:
2630:
2609:
2589:
2586:
2566:
2546:
2526:
2502:
2479:
2459:
2425:
2405:
2385:
2382:
2362:
2333:uniform spaces
2312:
2309:
2297:
2294:
2291:
2288:
2285:
2265:
2262:
2259:
2256:
2236:
2233:
2230:
2227:
2207:
2187:
2184:
2181:
2161:
2141:
2138:
2135:
2132:
2129:
2109:
2089:
2070:
2067:
2064:
2061:
2041:
2017:
2014:
2011:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1968:
1964:
1960:
1956:
1952:
1949:
1944:
1940:
1935:
1931:
1907:
1904:
1901:
1898:
1895:
1869:
1866:
1846:
1826:
1823:
1818:
1814:
1810:
1807:
1804:
1801:
1781:
1761:
1741:
1738:
1735:
1730:
1726:
1705:
1685:
1682:
1677:
1673:
1669:
1666:
1646:
1626:
1606:
1603:
1583:
1563:
1560:
1557:
1552:
1548:
1527:
1507:
1497:
1483:
1463:
1460:
1457:
1437:
1434:
1424:
1410:
1390:
1387:
1384:
1364:
1344:
1324:
1321:
1318:
1308:
1294:
1291:
1281:
1267:
1258:in which case
1247:
1244:
1220:
1200:
1197:
1177:
1157:
1137:
1134:
1131:
1120:
1119:
1103:
1100:
1080:
1060:
1040:
1020:
996:
993:
990:
987:
984:
964:
961:
956:
953:
950:
947:
944:
940:
936:
933:
913:
910:
907:
904:
901:
898:
875:
855:
835:
832:
827:
823:
819:
816:
796:
793:
790:
780:
777:is said to be
766:
746:
726:
723:
703:
683:
680:
648:
645:
625:
601:
581:
561:
531:
528:
525:
520:
516:
512:
509:
489:
469:
449:
446:
443:
438:
434:
430:
427:
407:
404:
401:
381:
378:
375:
370:
366:
345:
342:
318:
315:
312:
284:
281:
257:
237:
234:
231:
228:
225:
222:
213:; that is, if
202:
199:
196:
193:
190:
170:
167:
164:
153:
138:
135:
132:
129:
126:
103:
91:
88:
15:
9:
6:
4:
3:
2:
4307:
4296:
4293:
4292:
4290:
4275:
4267:
4263:
4260:
4258:
4255:
4253:
4250:
4249:
4248:
4240:
4238:
4234:
4230:
4228:
4224:
4220:
4218:
4213:
4208:
4206:
4198:
4197:
4194:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4159:
4157:
4155:
4151:
4145:
4144:Orientability
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4121:
4119:
4115:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4072:
4069:
4067:
4064:
4063:
4062:
4059:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4031:
4030:
4027:
4025:
4022:
4020:
4017:
4015:
4011:
4008:
4007:
4005:
4001:
3996:
3986:
3983:
3981:
3980:Set-theoretic
3978:
3974:
3971:
3970:
3969:
3966:
3962:
3959:
3958:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3941:Combinatorial
3939:
3937:
3934:
3932:
3929:
3928:
3926:
3922:
3918:
3911:
3906:
3904:
3899:
3897:
3892:
3891:
3888:
3880:
3876:
3872:
3866:
3862:
3858:
3857:Mineola, N.Y.
3854:
3853:
3847:
3843:
3839:
3835:
3829:
3825:
3821:
3817:
3813:
3809:
3805:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3769:
3765:
3761:
3757:
3756:
3748:
3746:0-13-181629-2
3742:
3738:
3737:Prentice Hall
3734:
3730:
3724:
3716:
3714:0-07-054235-X
3710:
3706:
3701:
3700:
3694:
3693:Rudin, Walter
3688:
3684:
3655:
3649:
3646:
3626:
3623:
3600:
3594:
3591:
3584:
3568:
3548:
3538:
3534:
3524:
3521:
3515:
3512:
3509:
3508:Neighbourhood
3506:
3503:
3500:
3494:
3493:Closed region
3491:
3485:
3482:
3479:
3476:
3475:
3455:
3452:
3432:
3412:
3392:
3386:
3383:
3380:
3372:
3348:
3344:
3341:
3337:
3329:
3326:
3322:
3301:
3298:
3295:
3285:
3281:
3278:
3274:
3256:
3253:
3250:
3240:
3236:
3216:
3210:
3207:
3204:
3181:
3161:
3153:
3132:
3126:
3123:
3120:
3094:
3091:
3088:
3078:
3077:unit interval
3074:
3070:
3067:
3065:
3046:
3043:
3040:
3030:
3026:
3025:
3019:
3017:
3016:
3008:
3003:
2989:
2969:
2966:
2958:
2942:
2922:
2919:
2899:
2891:
2875:
2847:
2844:
2841:
2818:
2798:
2756:
2733:
2708:
2696:
2693:
2689:
2684:
2681:
2679:
2675:
2672:
2668:
2667:
2666:
2653:
2645:
2639:
2629:
2627:
2624:if it has an
2623:
2607:
2600:Furthermore,
2587:
2584:
2564:
2544:
2524:
2516:
2500:
2491:
2477:
2457:
2448:
2445:
2443:
2439:
2423:
2403:
2383:
2380:
2360:
2352:
2348:
2345:
2340:
2338:
2334:
2330:
2326:
2325:metric spaces
2322:
2318:
2308:
2295:
2289:
2283:
2260:
2254:
2234:
2231:
2228:
2225:
2205:
2185:
2182:
2179:
2159:
2139:
2133:
2127:
2107:
2087:
2068:
2065:
2062:
2059:
2039:
2031:
2030:plain English
2015:
2012:
2009:
1983:
1977:
1971:
1966:
1962:
1958:
1954:
1950:
1947:
1942:
1938:
1933:
1929:
1921:
1905:
1899:
1896:
1893:
1885:
1880:
1867:
1864:
1844:
1824:
1821:
1816:
1812:
1808:
1805:
1802:
1799:
1779:
1759:
1739:
1736:
1733:
1728:
1724:
1703:
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1680:
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1671:
1667:
1664:
1644:
1624:
1604:
1601:
1581:
1561:
1558:
1555:
1550:
1546:
1525:
1505:
1495:
1481:
1461:
1458:
1455:
1435:
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1385:
1382:
1362:
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1322:
1316:
1306:
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1289:
1279:
1265:
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1242:
1234:
1218:
1198:
1195:
1175:
1155:
1135:
1132:
1129:
1117:
1116:
1115:
1101:
1098:
1078:
1058:
1038:
1018:
1010:
991:
985:
982:
962:
959:
951:
945:
942:
938:
934:
931:
911:
905:
899:
896:
889:
873:
853:
833:
830:
825:
821:
817:
814:
794:
791:
788:
778:
764:
744:
724:
721:
701:
681:
678:
670:
666:
662:
646:
643:
623:
615:
599:
592:is closed in
579:
559:
551:
547:
542:
529:
526:
523:
518:
514:
510:
507:
487:
467:
447:
444:
441:
436:
432:
428:
425:
405:
402:
399:
379:
376:
373:
368:
364:
343:
340:
332:
316:
313:
310:
302:
298:
282:
279:
271:
255:
235:
232:
229:
226:
220:
197:
194:
191:
168:
162:
154:
151:
133:
130:
127:
117:
101:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
30:
26:
22:
4274:Publications
4139:Chern number
4129:Betti number
4013:
4012: /
4003:Key concepts
3951:Differential
3851:
3823:
3793:
3763:
3732:
3723:
3698:
3687:
3561:is close to
3537:
3064:real numbers
3010:
3004:
2700:
2671:intersection
2641:
2515:disconnected
2492:
2449:
2446:
2341:
2337:gauge spaces
2314:
2276:is close to
1881:
1278:is called a
1121:
552:. A subset
543:
392:that is, if
297:limit points
150:
93:
68:limit points
47:
33:
4237:Wikiversity
4154:Key results
3705:McGraw-Hill
3345:The set of
3277:clopen sets
3027:The closed
2912:in a space
2152:Similarly,
1305:then there
44:mathematics
4083:CW complex
4024:Continuity
4014:Closed set
3973:cohomology
3679:References
3484:Closed map
3478:Clopen set
3325:Cantor set
3320:is closed.
2955:that is a
2694:is closed.
2636:See also:
2632:Properties
2626:open basis
1920:continuous
460:Moreover,
149:is called
78:under the
56:complement
48:closed set
4262:geometric
4257:algebraic
4108:Cobordism
4044:Hausdorff
4039:connected
3956:Geometric
3946:Continuum
3936:Algebraic
3842:175294365
3812:395340485
3782:945169917
3650:∪
3595:∪
3390:→
3305:∞
3217:∩
3133:∩
3007:countably
2892:of a set
2848:τ
2799:τ
2737:∅
2734:≠
2692:empty set
2317:open sets
2229:⊆
2183:∈
2063:⊆
2013:⊆
1972:
1959:⊆
1948:
1903:→
1822:
1809:∩
1752:However,
1734:
1681:
1556:
1459:⊆
1386:⊆
1320:∖
1133:∈
986:∪
960:
946:∪
935:∈
900:∪
831:
818:∈
792:⊆
781:a subset
665:sequences
546:sequences
524:
442:
429:⊆
403:⊆
374:
314:⊆
233:τ
230:∈
224:∖
198:τ
166:∖
134:τ
4289:Category
4227:Wikibook
4205:Category
4093:Manifold
4061:Homotopy
4019:Interior
4010:Open set
3968:Homology
3917:Topology
3822:(1996).
3794:Topology
3792:(1966).
3733:Topology
3731:(2000).
3695:(1976).
3583:subspace
3502:Open set
3472:See also
3347:integers
3239:interval
3154:between
3029:interval
3022:Examples
2957:superset
2683:finitely
2644:boundary
1886:: a map
924:meaning
779:close to
737:A point
60:open set
40:topology
36:geometry
21:open set
4252:general
4054:uniform
4034:compact
3985:Digital
2890:closure
2344:compact
1474:and if
1421:but to
886:in the
270:closure
70:. In a
62:. In a
4247:Topics
4049:metric
3924:Fields
3879:115240
3877:
3867:
3840:
3830:
3810:
3800:
3780:
3770:
3743:
3711:
3336:spaces
2335:, and
975:where
152:closed
76:closed
58:is an
54:whose
4029:Space
3529:Notes
2678:union
2396:then
2349:are "
2247:then
1518:then
1307:might
1231:is a
659:In a
614:limit
418:then
114:of a
80:limit
50:is a
3875:OCLC
3865:ISBN
3838:OCLC
3828:ISBN
3808:OCLC
3798:ISBN
3778:OCLC
3768:ISBN
3741:ISBN
3709:ISBN
3338:and
3323:The
3282:The
3174:and
3075:The
2690:The
2685:many
2676:The
2669:Any
2537:and
2032:as:
550:nets
548:and
46:, a
3373:If
3284:ray
3150:of
3062:of
2959:of
2811:on
2680:of
2620:is
2557:of
2513:is
2339:.
1918:is
1857:of
1594:in
1496:any
1494:is
1448:If
1423:not
1282:of
1211:If
1051:in
807:if
757:in
333:in
272:in
52:set
34:In
4291::
3873:.
3863:.
3859::
3855:.
3836:.
3806:.
3776:.
3739:.
3707:.
3703:.
2654:2.
2331:,
2327:,
1963:cl
1939:cl
1813:cl
1725:cl
1672:cl
1547:cl
939:cl
822:cl
515:cl
433:cl
365:cl
86:.
38:,
3909:e
3902:t
3895:v
3881:.
3844:.
3814:.
3784:.
3749:.
3717:.
3659:}
3656:x
3653:{
3647:A
3627:,
3624:X
3604:}
3601:x
3598:{
3592:A
3569:A
3549:x
3456:.
3453:X
3433:Y
3413:f
3393:Y
3387:X
3384::
3381:f
3357:Z
3342:.
3334:1
3332:T
3308:)
3302:+
3299:,
3296:1
3293:[
3279:.
3260:)
3257:1
3254:,
3251:0
3248:[
3221:Q
3214:]
3211:1
3208:,
3205:0
3202:[
3182:1
3162:0
3137:Q
3130:]
3127:1
3124:,
3121:0
3118:[
3098:]
3095:1
3092:,
3089:0
3086:[
3050:]
3047:b
3044:,
3041:a
3038:[
3014:σ
3012:F
2990:X
2970:.
2967:A
2943:X
2923:,
2920:X
2900:A
2876:.
2872:F
2851:)
2845:,
2842:X
2839:(
2819:X
2778:F
2757:X
2730:F
2709:X
2608:X
2588:.
2585:X
2565:X
2545:B
2525:A
2501:X
2478:X
2458:X
2424:X
2404:D
2384:,
2381:X
2361:D
2296:.
2293:)
2290:A
2287:(
2284:f
2264:)
2261:x
2258:(
2255:f
2235:,
2232:X
2226:A
2206:x
2186:X
2180:x
2160:f
2140:.
2137:)
2134:A
2131:(
2128:f
2108:A
2088:f
2069:,
2066:X
2060:A
2040:f
2016:X
2010:A
1990:)
1987:)
1984:A
1981:(
1978:f
1975:(
1967:Y
1955:)
1951:A
1943:X
1934:(
1930:f
1906:Y
1900:X
1897::
1894:f
1868:.
1865:X
1845:Y
1825:A
1817:Y
1806:X
1803:=
1800:A
1780:X
1760:A
1740:.
1737:A
1729:Y
1704:A
1684:A
1676:X
1668:=
1665:A
1645:X
1625:A
1605:;
1602:Y
1582:A
1562:,
1559:A
1551:Y
1526:A
1506:X
1482:Y
1462:X
1456:A
1436:.
1433:Y
1409:X
1389:X
1383:A
1363:X
1343:A
1323:X
1317:Y
1293:,
1290:X
1266:Y
1246:,
1243:Y
1219:X
1199:.
1196:x
1176:A
1156:A
1136:X
1130:x
1102:,
1099:A
1079:X
1059:X
1039:A
1019:X
995:}
992:x
989:{
983:A
963:A
955:}
952:x
949:{
943:A
932:x
912:,
909:}
906:x
903:{
897:A
874:A
854:x
834:A
826:X
815:x
795:X
789:A
765:X
745:x
725:.
722:X
702:X
682:,
679:X
647:.
644:A
624:A
600:X
580:X
560:A
530:.
527:A
519:X
511:=
508:A
488:X
468:A
448:.
445:A
437:X
426:A
406:X
400:A
380:;
377:A
369:X
344:,
341:X
317:X
311:A
283:.
280:X
256:X
236:.
227:A
221:X
201:)
195:,
192:X
189:(
169:A
163:X
137:)
131:,
128:X
125:(
102:A
31:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.