4071:
3854:
4092:
4060:
4129:
4102:
4082:
1609:
3128:
3046:
2886:
2483:
1676:
1449:
1080:
3358:
2750:
2668:
3391:
2350:
2245:
2169:
1816:
1792:
1286:
3427:
3302:
1718:
1165:
877:
756:
615:
1641:
1475:
1380:
3269:
3169:, and the only closed points are the maximal ideals, and are thus not contained in any of the open sets of the topology, and thus the space does not satisfy axiom T
2590:
2559:
2421:
2394:
2314:
2022:
1331:
1195:
1106:
641:
2946:
3454:
3238:
2801:
2212:
2089:
1761:
1218:
967:
806:
668:
4132:
3211:
3191:
2966:
2920:
2821:
2770:
2688:
2610:
2532:
2285:
2265:
2189:
2129:
2109:
2066:
2042:
1988:
1738:
1562:
1542:
1522:
1499:
1406:
1351:
1305:
1244:
1129:
1048:
1024:
993:
944:
920:
896:
848:
828:
780:
724:
690:
586:
551:
525:
502:
3456:
Notice how this example differs subtly from the cofinite topology example, above: the points in the topology are not closed, in general, whereas in a T
4153:
125:
107:
3693:
3766:
4120:
4115:
3735:
3719:
3671:
3647:
2499:
239:
1567:
4110:
3472:
4012:
1925:
378:
283:
430:
3504:. The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant
3058:
2995:
1614:
341:
271:
2832:
4158:
4020:
2982:
2426:
2777:
2366:
4091:
3819:
1646:
923:
2968:
even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.
1422:
1053:
4105:
3307:
1385:
4040:
4035:
3961:
3838:
3826:
3799:
3759:
1970:). This follows since no two nonempty open sets of the cofinite topology are disjoint. Specifically, let
1940:
1478:
2697:
2615:
3882:
3809:
3363:
1890:
4070:
464:(i.e., a space in which distinct points are topologically distinguishable). A topological space is an R
4030:
3982:
3956:
3804:
3548:
3536:
3464:
1413:
899:
4081:
3877:
3663:
1906:
1223:
999:
972:
469:
418:
4075:
4025:
3946:
3936:
3814:
3794:
3146:
1409:
1168:
38:
4045:
2319:
2217:
2141:
1799:
1775:
1256:
4063:
3929:
3887:
3752:
3731:
3715:
3699:
3689:
3667:
3643:
3501:
3396:
2990:
2978:
1951:
1766:
In any topological space we have, as properties of any two points, the following implications
782:
759:
311:
267:
45:
3278:
1694:
1141:
853:
732:
591:
3843:
3789:
3655:
3142:
2974:
2773:
1620:
1454:
1356:
1003:
554:
457:
434:
414:
295:
53:
33:
3247:
2568:
2537:
2399:
2372:
2293:
2000:
1310:
1174:
1085:
620:
3902:
3897:
3241:
3131:
1963:
1894:
442:
161:
89:
3656:
2925:
3436:
3220:
2783:
2194:
2071:
1743:
1200:
949:
788:
650:
3992:
3924:
3681:
3505:
3196:
3176:
3130:
Thus, the point is closed. However, this example is well known as a space that is not
2951:
2905:
2806:
2755:
2673:
2595:
2517:
2270:
2250:
2174:
2114:
2094:
2051:
2027:
1973:
1859:
1723:
1547:
1527:
1507:
1484:
1391:
1336:
1290:
1229:
1114:
1033:
1009:
978:
929:
905:
881:
833:
813:
765:
709:
675:
571:
536:
510:
487:
336:
3492:", and their synonyms can also be applied to such variations of topological spaces as
4147:
4002:
3912:
3892:
3493:
3166:
143:
4095:
3714:. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.
3535:
alone can be an interesting condition on other sorts of convergence spaces, such as
3987:
3907:
3853:
3605:
3497:
1955:
216:
196:
178:
4085:
3997:
3214:
3158:
2691:
2486:
255:
3519:
As it turns out, uniform spaces, and more generally Cauchy spaces, are always R
3173:. To be clear about this example: the Zariski topology for a commutative ring
3941:
3872:
3831:
3053:
2490:
2045:
1685:
696:
644:
3703:
3966:
1050:
is a topological space then the following conditions are equivalent: (where
3157:. To see this, note that the closure of a one-point set is the set of all
1740:
converges only to the points that are topologically indistinguishable from
3429:
The closed sets of the
Zariski topology are the sets of prime ideals that
3138:). The Zariski topology is essentially an example of a cofinite topology.
3951:
3919:
3868:
3775:
3049:
1995:
1829:
1681:
703:
251:
17:
2562:
1991:
1197:
contains only the points that are topologically indistinguishable from
3304:
It is straightforward to verify that this indeed forms the basis: so
504:
is a topological space then the following conditions are equivalent:
2485:
which is never empty. Alternatively, the set of even integers is
2989:. To see this, note that the singleton containing a point with
3748:
3512:
spaces) or unique up to topological indistinguishability (for R
1885:. In contrast, the corresponding statement does not hold for T
1501:(that is, any two such sets are either identical or disjoint).
3744:
1873:
neighbourhood (when given the subspace topology), is also T
2534:
be the set of integers again, and using the definition of
1604:{\displaystyle F\cap \operatorname {cl} \{x\}=\emptyset .}
2498:
The above example can be modified slightly to create the
270:
in which, for every pair of distinct points, each has a
27:
Topological space in which all singleton sets are closed
3658:
Real analysis: modern techniques and their applications
1835:. If the composite arrow can be reversed the space is T
692:
is the intersection of all the open sets containing it.
3193:
is given as follows: the topological space is the set
2361:
by each of the definitions above. This space is not T
3439:
3399:
3366:
3310:
3281:
3250:
3223:
3199:
3179:
3061:
2998:
2954:
2928:
2908:
2835:
2809:
2786:
2758:
2700:
2676:
2618:
2598:
2571:
2540:
2520:
2429:
2402:
2375:
2322:
2296:
2273:
2253:
2220:
2197:
2177:
2144:
2117:
2097:
2074:
2054:
2030:
2003:
1976:
1802:
1778:
1746:
1726:
1697:
1649:
1623:
1570:
1550:
1530:
1510:
1487:
1457:
1425:
1394:
1359:
1339:
1313:
1293:
1259:
1232:
1203:
1177:
1144:
1117:
1088:
1056:
1036:
1012:
981:
952:
932:
908:
884:
856:
836:
816:
791:
768:
735:
712:
678:
653:
623:
594:
574:
539:
513:
490:
3662:(2nd ed.). John Wiley & Sons, Inc. p.
4011:
3975:
3861:
3782:
1828:. If the second arrow can be reversed the space is
215:
195:
177:
160:
142:
124:
106:
88:
70:
52:
44:
32:
3448:
3421:
3385:
3352:
3296:
3263:
3232:
3205:
3185:
3161:that contain the point (and thus the topology is T
3122:
3040:
2960:
2940:
2914:
2880:
2815:
2795:
2764:
2744:
2682:
2662:
2604:
2584:
2553:
2526:
2477:
2415:
2388:
2344:
2308:
2279:
2259:
2239:
2206:
2183:
2163:
2123:
2103:
2083:
2060:
2036:
2016:
1982:
1810:
1786:
1755:
1732:
1712:
1670:
1635:
1603:
1556:
1536:
1516:
1493:
1469:
1443:
1400:
1374:
1345:
1325:
1299:
1280:
1238:
1212:
1189:
1159:
1123:
1100:
1074:
1042:
1018:
987:
961:
938:
914:
890:
871:
842:
822:
800:
774:
750:
718:
684:
662:
635:
609:
580:
545:
519:
496:
2493:, which would be impossible in a Hausdorff space.
1824:If the first arrow can be reversed the space is R
3123:{\displaystyle x_{1}-c_{1},\ldots ,x_{n}-c_{n}.}
3041:{\displaystyle \left(c_{1},\ldots ,c_{n}\right)}
3710:Lynn Arthur Steen and J. Arthur Seebach, Jr.,
2881:{\displaystyle U_{A}:=\bigcap _{x\in A}G_{x}.}
3760:
282:is one in which this holds for every pair of
8:
3638:A.V. Arkhangel'skii, L.S. Pontryagin (Eds.)
2737:
2719:
2655:
2637:
2478:{\displaystyle O_{A}\cap O_{B}=O_{A\cup B},}
2334:
2328:
2303:
2297:
2232:
2226:
2156:
2150:
1662:
1656:
1589:
1583:
1438:
1432:
1366:
1360:
1320:
1314:
1184:
1178:
1095:
1089:
1069:
1063:
630:
624:
1958:is a simple example of a topology that is T
1928:is a simple example of a topology that is T
1909:is a simple example of a topology that is T
4128:
4101:
3767:
3753:
3745:
3606:"Locally Euclidean space implies T1 space"
1807:
1803:
1783:
1779:
429:is preferred. There is also a notion of a
3592:
3527:condition in these cases reduces to the T
3438:
3404:
3398:
3371:
3365:
3341:
3328:
3315:
3309:
3280:
3255:
3249:
3222:
3198:
3178:
3111:
3098:
3079:
3066:
3060:
3027:
3008:
2997:
2953:
2927:
2907:
2869:
2853:
2840:
2834:
2808:
2785:
2780:of the subbasic sets: given a finite set
2757:
2718:
2705:
2699:
2675:
2636:
2623:
2617:
2597:
2576:
2570:
2545:
2539:
2519:
2460:
2447:
2434:
2428:
2407:
2401:
2380:
2374:
2327:
2321:
2295:
2272:
2252:
2225:
2219:
2196:
2176:
2149:
2143:
2116:
2096:
2073:
2053:
2029:
2008:
2002:
1975:
1801:
1777:
1745:
1725:
1696:
1671:{\displaystyle \operatorname {cl} \{x\}.}
1648:
1622:
1569:
1549:
1529:
1509:
1486:
1456:
1424:
1393:
1358:
1338:
1312:
1292:
1258:
1231:
1202:
1176:
1143:
1116:
1087:
1055:
1035:
1011:
980:
951:
931:
907:
883:
855:
835:
815:
790:
767:
734:
711:
677:
652:
622:
593:
573:
538:
512:
489:
3686:Handbook of Analysis and Its Foundations
3551: – Mathematical property of a space
3480:Generalisations to other kinds of spaces
1947:but the converse is not true in general.
1444:{\displaystyle \operatorname {cl} \{x\}}
1075:{\displaystyle \operatorname {cl} \{x\}}
3561:
3380:
1877:. Similarly, a space that is locally R
344:that does not contain the other point.
3353:{\displaystyle O_{a}\cap O_{b}=O_{ab}}
1869:, in the sense that each point has a T
29:
7:
2776:of the topology are given by finite
2561:from the previous example, define a
452:space if and only if it is both an R
274:not containing the other point. An
3730:. New York: Dover. pp. 86–90.
2745:{\displaystyle G_{x}=O_{\{x-1,x\}}}
2663:{\displaystyle G_{x}=O_{\{x,x+1\}}}
946:contains infinitely many points of
3386:{\displaystyle O_{0}=\varnothing }
2316:is the complement of the open set
2290:equivalently, every singleton set
1595:
25:
3688:. San Diego, CA: Academic Press.
4154:Properties of topological spaces
4127:
4100:
4090:
4080:
4069:
4059:
4058:
3852:
3582:See proposition 13, section 2.6.
3460:space, points are always closed.
2500:double-pointed cofinite topology
3165:). However, this closure is a
2502:, which is an example of an R
1804:
1780:
359:if any two distinct points in
1:
2091:Then given distinct integers
1926:overlapping interval topology
1862:(since every set is closed).
1795:topologically distinguishable
1006:with respect to the map from
379:topologically distinguishable
284:topologically distinguishable
3624:See example 21, section 2.6.
2894:The resulting space is not T
1002:to the single point has the
421:. For this reason, the term
3712:Counterexamples in Topology
2357:so the resulting space is T
1843:if and only if it is both R
588:; that is, for every point
4175:
4021:Banach fixed-point theorem
3610:Mathematics Stack Exchange
3244:is given by the open sets
3141:The Zariski topology on a
2983:algebraically closed field
2345:{\displaystyle O_{\{x\}},}
1897:but is locally Hausdorff.
1889:spaces. For example, the
922:if and only if every open
448:A topological space is a T
415:entirely different meaning
4054:
3850:
3726:Willard, Stephen (1998).
3471:, since every point is a
2240:{\displaystyle O_{\{y\}}}
2164:{\displaystyle O_{\{x\}}}
1865:A space that is locally T
1811:{\displaystyle \implies }
1787:{\displaystyle \implies }
1720:the fixed ultrafilter at
1281:{\displaystyle x,y\in X,}
468:space if and only if its
286:points. The properties T
235:
3654:Folland, Gerald (1999).
3422:{\displaystyle O_{1}=X.}
3271:of prime ideals that do
393:space is also called an
254:and related branches of
3642:(1990) Springer-Verlag
3622:Arkhangel'skii (1990).
3568:Arkhangel'skii (1990).
3297:{\displaystyle a\in A.}
2506:space that is neither T
2044:that contain all but a
2024:to be those subsets of
1713:{\displaystyle x\in X,}
1386:specialization preorder
1160:{\displaystyle x\in X,}
1082:denotes the closure of
872:{\displaystyle x\in X,}
751:{\displaystyle x\in X,}
610:{\displaystyle x\in X,}
405:space is also called a
4076:Mathematics portal
3976:Metrics and properties
3962:Second-countable space
3580:Archangel'skii (1990)
3450:
3423:
3387:
3354:
3298:
3265:
3234:
3207:
3187:
3153:but not, in general, T
3124:
3042:
2962:
2942:
2916:
2902:), because the points
2882:
2817:
2797:
2766:
2746:
2684:
2664:
2606:
2586:
2555:
2528:
2479:
2417:
2390:
2352:so it is a closed set;
2346:
2310:
2281:
2261:
2241:
2208:
2185:
2165:
2125:
2105:
2085:
2062:
2038:
2018:
1984:
1941:weakly Hausdorff space
1917:, and hence also not R
1812:
1788:
1757:
1734:
1714:
1672:
1637:
1636:{\displaystyle x\in X}
1605:
1558:
1538:
1518:
1495:
1471:
1470:{\displaystyle x\in X}
1445:
1402:
1376:
1375:{\displaystyle \{x\}.}
1347:
1327:
1301:
1282:
1240:
1214:
1191:
1161:
1125:
1102:
1076:
1044:
1020:
989:
963:
940:
916:
892:
873:
844:
824:
802:
776:
752:
720:
686:
664:
637:
611:
582:
547:
521:
498:
138:(completely Hausdorff)
3537:pretopological spaces
3475:and therefore closed.
3451:
3424:
3388:
3355:
3299:
3266:
3264:{\displaystyle O_{a}}
3235:
3208:
3188:
3125:
3043:
2963:
2943:
2917:
2883:
2818:
2798:
2767:
2747:
2685:
2665:
2607:
2587:
2585:{\displaystyle G_{x}}
2556:
2554:{\displaystyle O_{A}}
2529:
2480:
2418:
2416:{\displaystyle O_{B}}
2391:
2389:{\displaystyle O_{A}}
2369:of any two open sets
2347:
2311:
2309:{\displaystyle \{x\}}
2282:
2262:
2242:
2209:
2186:
2166:
2126:
2106:
2086:
2063:
2039:
2019:
2017:{\displaystyle O_{A}}
1985:
1891:line with two origins
1858:space is necessarily
1813:
1789:
1758:
1735:
1715:
1673:
1638:
1606:
1559:
1539:
1519:
1496:
1472:
1446:
1403:
1377:
1353:is in the closure of
1348:
1328:
1326:{\displaystyle \{y\}}
1307:is in the closure of
1302:
1283:
1241:
1215:
1192:
1190:{\displaystyle \{x\}}
1162:
1126:
1103:
1101:{\displaystyle \{x\}}
1077:
1045:
1021:
990:
964:
941:
917:
893:
874:
845:
825:
803:
777:
753:
721:
687:
665:
638:
636:{\displaystyle \{x\}}
612:
583:
568:Points are closed in
548:
522:
499:
431:Fréchet–Urysohn space
4031:Invariance of domain
3983:Euler characteristic
3957:Bundle (mathematics)
3549:Topological property
3465:totally disconnected
3437:
3397:
3364:
3308:
3279:
3248:
3242:base of the topology
3221:
3197:
3177:
3145:(that is, the prime
3059:
2996:
2952:
2926:
2906:
2833:
2807:
2784:
2756:
2698:
2674:
2616:
2596:
2569:
2538:
2518:
2427:
2400:
2373:
2320:
2294:
2271:
2251:
2218:
2195:
2175:
2142:
2115:
2095:
2072:
2052:
2028:
2001:
1974:
1800:
1776:
1744:
1724:
1695:
1647:
1621:
1568:
1548:
1528:
1524:is a closed set and
1508:
1485:
1455:
1423:
1414:equivalence relation
1392:
1357:
1337:
1311:
1291:
1257:
1230:
1201:
1175:
1142:
1115:
1086:
1054:
1034:
1026:to the single point.
1010:
979:
950:
930:
906:
882:
854:
834:
814:
789:
766:
733:
710:
676:
651:
621:
592:
572:
537:
511:
488:
4041:Tychonoff's theorem
4036:Poincaré conjecture
3790:General (point-set)
3508:) are unique (for T
3473:connected component
2941:{\displaystyle x+1}
1224:Kolmogorov quotient
470:Kolmogorov quotient
419:functional analysis
156:(regular Hausdorff)
4026:De Rham cohomology
3947:Polyhedral complex
3937:Simplicial complex
3640:General Topology I
3502:convergence spaces
3449:{\displaystyle a.}
3446:
3419:
3383:
3350:
3294:
3261:
3233:{\displaystyle A.}
3230:
3203:
3183:
3147:spectrum of a ring
3120:
3038:
2958:
2938:
2912:
2878:
2864:
2813:
2796:{\displaystyle A,}
2793:
2772:is odd. Then the
2762:
2742:
2680:
2660:
2602:
2582:
2551:
2524:
2475:
2413:
2386:
2342:
2306:
2277:
2257:
2237:
2207:{\displaystyle x,}
2204:
2181:
2161:
2121:
2101:
2084:{\displaystyle X.}
2081:
2058:
2034:
2014:
1980:
1808:
1784:
1756:{\displaystyle x.}
1753:
1730:
1710:
1668:
1633:
1601:
1554:
1544:is a point not in
1534:
1514:
1491:
1467:
1441:
1412:(and therefore an
1398:
1372:
1343:
1323:
1297:
1278:
1236:
1213:{\displaystyle x.}
1210:
1187:
1157:
1121:
1098:
1072:
1040:
1016:
985:
971:Each map from the
962:{\displaystyle S.}
959:
936:
912:
888:
869:
840:
820:
801:{\displaystyle x.}
798:
772:
748:
716:
682:
663:{\displaystyle X.}
660:
633:
617:the singleton set
607:
578:
543:
517:
494:
340:if each lies in a
209:(completely normal
191:(normal Hausdorff)
39:topological spaces
4159:Separation axioms
4141:
4140:
3930:fundamental group
3695:978-0-12-622760-4
3206:{\displaystyle X}
3186:{\displaystyle A}
2991:local coordinates
2979:algebraic variety
2961:{\displaystyle x}
2915:{\displaystyle x}
2849:
2816:{\displaystyle X}
2803:the open sets of
2765:{\displaystyle x}
2683:{\displaystyle x}
2605:{\displaystyle x}
2527:{\displaystyle X}
2280:{\displaystyle y}
2260:{\displaystyle x}
2214:and the open set
2184:{\displaystyle y}
2124:{\displaystyle y}
2104:{\displaystyle x}
2061:{\displaystyle A}
2037:{\displaystyle X}
1994:, and define the
1983:{\displaystyle X}
1952:cofinite topology
1733:{\displaystyle x}
1557:{\displaystyle F}
1537:{\displaystyle x}
1517:{\displaystyle F}
1494:{\displaystyle X}
1401:{\displaystyle X}
1346:{\displaystyle y}
1300:{\displaystyle x}
1239:{\displaystyle X}
1124:{\displaystyle X}
1043:{\displaystyle X}
1019:{\displaystyle X}
998:The map from the
988:{\displaystyle X}
939:{\displaystyle x}
915:{\displaystyle S}
891:{\displaystyle x}
843:{\displaystyle X}
823:{\displaystyle S}
810:For every subset
775:{\displaystyle x}
760:fixed ultrafilter
719:{\displaystyle X}
685:{\displaystyle X}
581:{\displaystyle X}
546:{\displaystyle X}
520:{\displaystyle X}
497:{\displaystyle X}
312:topological space
296:separation axioms
268:topological space
248:
247:
229:(perfectly normal
34:Separation axioms
16:(Redirected from
4166:
4131:
4130:
4104:
4103:
4094:
4084:
4074:
4073:
4062:
4061:
3856:
3769:
3762:
3755:
3746:
3741:
3728:General Topology
3722:(Dover edition).
3707:
3677:
3661:
3626:
3620:
3614:
3613:
3602:
3596:
3590:
3584:
3578:
3572:
3570:See section 2.6.
3566:
3531:condition. But R
3455:
3453:
3452:
3447:
3428:
3426:
3425:
3420:
3409:
3408:
3392:
3390:
3389:
3384:
3376:
3375:
3359:
3357:
3356:
3351:
3349:
3348:
3333:
3332:
3320:
3319:
3303:
3301:
3300:
3295:
3270:
3268:
3267:
3262:
3260:
3259:
3239:
3237:
3236:
3231:
3212:
3210:
3209:
3204:
3192:
3190:
3189:
3184:
3143:commutative ring
3129:
3127:
3126:
3121:
3116:
3115:
3103:
3102:
3084:
3083:
3071:
3070:
3047:
3045:
3044:
3039:
3037:
3033:
3032:
3031:
3013:
3012:
2975:Zariski topology
2967:
2965:
2964:
2959:
2947:
2945:
2944:
2939:
2921:
2919:
2918:
2913:
2898:(and hence not T
2887:
2885:
2884:
2879:
2874:
2873:
2863:
2845:
2844:
2822:
2820:
2819:
2814:
2802:
2800:
2799:
2794:
2771:
2769:
2768:
2763:
2751:
2749:
2748:
2743:
2741:
2740:
2710:
2709:
2689:
2687:
2686:
2681:
2669:
2667:
2666:
2661:
2659:
2658:
2628:
2627:
2611:
2609:
2608:
2603:
2592:for any integer
2591:
2589:
2588:
2583:
2581:
2580:
2560:
2558:
2557:
2552:
2550:
2549:
2533:
2531:
2530:
2525:
2484:
2482:
2481:
2476:
2471:
2470:
2452:
2451:
2439:
2438:
2422:
2420:
2419:
2414:
2412:
2411:
2395:
2393:
2392:
2387:
2385:
2384:
2351:
2349:
2348:
2343:
2338:
2337:
2315:
2313:
2312:
2307:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2236:
2235:
2213:
2211:
2210:
2205:
2190:
2188:
2187:
2182:
2170:
2168:
2167:
2162:
2160:
2159:
2130:
2128:
2127:
2122:
2110:
2108:
2107:
2102:
2090:
2088:
2087:
2082:
2067:
2065:
2064:
2059:
2043:
2041:
2040:
2035:
2023:
2021:
2020:
2015:
2013:
2012:
1989:
1987:
1986:
1981:
1907:Sierpiński space
1817:
1815:
1814:
1809:
1793:
1791:
1790:
1785:
1762:
1760:
1759:
1754:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1677:
1675:
1674:
1669:
1642:
1640:
1639:
1634:
1610:
1608:
1607:
1602:
1563:
1561:
1560:
1555:
1543:
1541:
1540:
1535:
1523:
1521:
1520:
1515:
1500:
1498:
1497:
1492:
1476:
1474:
1473:
1468:
1450:
1448:
1447:
1442:
1407:
1405:
1404:
1399:
1381:
1379:
1378:
1373:
1352:
1350:
1349:
1344:
1332:
1330:
1329:
1324:
1306:
1304:
1303:
1298:
1287:
1285:
1284:
1279:
1245:
1243:
1242:
1237:
1219:
1217:
1216:
1211:
1196:
1194:
1193:
1188:
1166:
1164:
1163:
1158:
1130:
1128:
1127:
1122:
1107:
1105:
1104:
1099:
1081:
1079:
1078:
1073:
1049:
1047:
1046:
1041:
1025:
1023:
1022:
1017:
1004:lifting property
1000:Sierpiński space
994:
992:
991:
986:
973:Sierpiński space
968:
966:
965:
960:
945:
943:
942:
937:
921:
919:
918:
913:
897:
895:
894:
889:
878:
876:
875:
870:
850:and every point
849:
847:
846:
841:
829:
827:
826:
821:
807:
805:
804:
799:
781:
779:
778:
773:
757:
755:
754:
749:
725:
723:
722:
717:
691:
689:
688:
683:
672:Every subset of
669:
667:
666:
661:
642:
640:
639:
634:
616:
614:
613:
608:
587:
585:
584:
579:
552:
550:
549:
544:
526:
524:
523:
518:
503:
501:
500:
495:
458:Kolmogorov (or T
435:sequential space
399:Fréchet topology
397:or a space with
395:accessible space
294:are examples of
231: Hausdorff)
226:
221:
211: Hausdorff)
206:
201:
188:
183:
168:
167:
153:
148:
135:
130:
115:
114:
99:
94:
81:
76:
63:
58:
30:
21:
4174:
4173:
4169:
4168:
4167:
4165:
4164:
4163:
4144:
4143:
4142:
4137:
4068:
4050:
4046:Urysohn's lemma
4007:
3971:
3857:
3848:
3820:low-dimensional
3778:
3773:
3738:
3725:
3696:
3682:Schechter, Eric
3680:
3674:
3653:
3635:
3630:
3629:
3621:
3617:
3604:
3603:
3599:
3595:, 16.6, p. 438.
3591:
3587:
3579:
3575:
3567:
3563:
3558:
3545:
3534:
3530:
3526:
3522:
3515:
3511:
3491:
3487:
3482:
3470:
3459:
3435:
3434:
3400:
3395:
3394:
3367:
3362:
3361:
3337:
3324:
3311:
3306:
3305:
3277:
3276:
3251:
3246:
3245:
3219:
3218:
3195:
3194:
3175:
3174:
3172:
3164:
3156:
3152:
3137:
3107:
3094:
3075:
3062:
3057:
3056:
3023:
3004:
3003:
2999:
2994:
2993:
2988:
2950:
2949:
2924:
2923:
2904:
2903:
2901:
2897:
2865:
2836:
2831:
2830:
2805:
2804:
2782:
2781:
2754:
2753:
2714:
2701:
2696:
2695:
2672:
2671:
2632:
2619:
2614:
2613:
2594:
2593:
2572:
2567:
2566:
2541:
2536:
2535:
2516:
2515:
2513:
2509:
2505:
2456:
2443:
2430:
2425:
2424:
2403:
2398:
2397:
2376:
2371:
2370:
2364:
2360:
2323:
2318:
2317:
2292:
2291:
2269:
2268:
2249:
2248:
2221:
2216:
2215:
2193:
2192:
2173:
2172:
2145:
2140:
2139:
2113:
2112:
2093:
2092:
2070:
2069:
2050:
2049:
2026:
2025:
2004:
1999:
1998:
1972:
1971:
1969:
1961:
1946:
1935:
1931:
1920:
1916:
1912:
1903:
1895:Hausdorff space
1888:
1884:
1880:
1876:
1872:
1868:
1857:
1850:
1846:
1842:
1838:
1833:
1827:
1798:
1797:
1774:
1773:
1742:
1741:
1722:
1721:
1693:
1692:
1645:
1644:
1619:
1618:
1566:
1565:
1546:
1545:
1526:
1525:
1506:
1505:
1483:
1482:
1453:
1452:
1421:
1420:
1390:
1389:
1355:
1354:
1335:
1334:
1333:if and only if
1309:
1308:
1289:
1288:
1255:
1254:
1249:
1228:
1227:
1199:
1198:
1173:
1172:
1140:
1139:
1134:
1113:
1112:
1084:
1083:
1052:
1051:
1032:
1031:
1008:
1007:
977:
976:
948:
947:
928:
927:
904:
903:
880:
879:
852:
851:
832:
831:
812:
811:
787:
786:
764:
763:
731:
730:
708:
707:
674:
673:
649:
648:
619:
618:
590:
589:
570:
569:
564:
558:
535:
534:
530:
509:
508:
486:
485:
482:
475:
467:
461:
455:
451:
443:another meaning
439:symmetric space
426:
407:symmetric space
404:
392:
374:
356:
304:
293:
289:
279:
263:
244:
230:
224:
222:
219:
210:
204:
202:
199:
186:
184:
181:
169:
165:
164:
151:
149:
146:
133:
131:
128:
116:
112:
110:
97:
95:
92:
79:
77:
74:
61:
59:
56:
36:
28:
23:
22:
15:
12:
11:
5:
4172:
4170:
4162:
4161:
4156:
4146:
4145:
4139:
4138:
4136:
4135:
4125:
4124:
4123:
4118:
4113:
4098:
4088:
4078:
4066:
4055:
4052:
4051:
4049:
4048:
4043:
4038:
4033:
4028:
4023:
4017:
4015:
4009:
4008:
4006:
4005:
4000:
3995:
3993:Winding number
3990:
3985:
3979:
3977:
3973:
3972:
3970:
3969:
3964:
3959:
3954:
3949:
3944:
3939:
3934:
3933:
3932:
3927:
3925:homotopy group
3917:
3916:
3915:
3910:
3905:
3900:
3895:
3885:
3880:
3875:
3865:
3863:
3859:
3858:
3851:
3849:
3847:
3846:
3841:
3836:
3835:
3834:
3824:
3823:
3822:
3812:
3807:
3802:
3797:
3792:
3786:
3784:
3780:
3779:
3774:
3772:
3771:
3764:
3757:
3749:
3743:
3742:
3736:
3723:
3708:
3694:
3678:
3672:
3651:
3634:
3631:
3628:
3627:
3615:
3597:
3593:Schechter 1996
3585:
3573:
3560:
3559:
3557:
3554:
3553:
3552:
3544:
3541:
3532:
3528:
3524:
3520:
3513:
3509:
3494:uniform spaces
3489:
3485:
3481:
3478:
3477:
3476:
3468:
3461:
3457:
3445:
3442:
3432:
3418:
3415:
3412:
3407:
3403:
3382:
3379:
3374:
3370:
3347:
3344:
3340:
3336:
3331:
3327:
3323:
3318:
3314:
3293:
3290:
3287:
3284:
3274:
3258:
3254:
3229:
3226:
3202:
3182:
3170:
3162:
3154:
3150:
3139:
3135:
3119:
3114:
3110:
3106:
3101:
3097:
3093:
3090:
3087:
3082:
3078:
3074:
3069:
3065:
3036:
3030:
3026:
3022:
3019:
3016:
3011:
3007:
3002:
2986:
2970:
2969:
2957:
2937:
2934:
2931:
2911:
2899:
2895:
2891:
2890:
2889:
2888:
2877:
2872:
2868:
2862:
2859:
2856:
2852:
2848:
2843:
2839:
2825:
2824:
2812:
2792:
2789:
2761:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2717:
2713:
2708:
2704:
2679:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2635:
2631:
2626:
2622:
2601:
2579:
2575:
2548:
2544:
2523:
2511:
2507:
2503:
2495:
2494:
2474:
2469:
2466:
2463:
2459:
2455:
2450:
2446:
2442:
2437:
2433:
2410:
2406:
2383:
2379:
2365:, because the
2362:
2358:
2355:
2354:
2353:
2341:
2336:
2333:
2330:
2326:
2305:
2302:
2299:
2288:
2276:
2256:
2234:
2231:
2228:
2224:
2203:
2200:
2180:
2158:
2155:
2152:
2148:
2133:
2132:
2120:
2100:
2080:
2077:
2057:
2033:
2011:
2007:
1990:be the set of
1979:
1967:
1959:
1948:
1944:
1937:
1933:
1929:
1922:
1918:
1914:
1910:
1902:
1899:
1886:
1882:
1878:
1874:
1870:
1866:
1855:
1848:
1844:
1840:
1839:. A space is T
1836:
1831:
1825:
1822:
1821:
1820:
1806:
1796:
1782:
1772:
1764:
1763:
1752:
1749:
1729:
1709:
1706:
1703:
1700:
1689:
1684:is a union of
1678:
1667:
1664:
1661:
1658:
1655:
1652:
1632:
1629:
1626:
1611:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1553:
1533:
1513:
1502:
1490:
1466:
1463:
1460:
1440:
1437:
1434:
1431:
1428:
1417:
1397:
1382:
1371:
1368:
1365:
1362:
1342:
1322:
1319:
1316:
1296:
1277:
1274:
1271:
1268:
1265:
1262:
1251:
1247:
1235:
1220:
1209:
1206:
1186:
1183:
1180:
1156:
1153:
1150:
1147:
1136:
1132:
1120:
1097:
1094:
1091:
1071:
1068:
1065:
1062:
1059:
1039:
1028:
1027:
1015:
996:
984:
969:
958:
955:
935:
911:
887:
868:
865:
862:
859:
839:
819:
808:
797:
794:
771:
747:
744:
741:
738:
727:
715:
700:
693:
681:
670:
659:
656:
632:
629:
626:
606:
603:
600:
597:
577:
566:
562:
556:
542:
532:
528:
516:
493:
481:
478:
473:
465:
459:
453:
449:
440:
424:
412:
402:
390:
387:
386:
385:are separated.
372:
364:
363:are separated.
354:
339:
326:. We say that
303:
300:
291:
287:
277:
261:
246:
245:
243:
242:
236:
233:
232:
227:
218:
213:
212:
207:
198:
193:
192:
189:
180:
175:
174:
171:
163:
158:
157:
154:
145:
140:
139:
136:
127:
122:
121:
118:
109:
104:
103:
100:
91:
86:
85:
82:
73:
68:
67:
64:
55:
50:
49:
48:classification
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4171:
4160:
4157:
4155:
4152:
4151:
4149:
4134:
4126:
4122:
4119:
4117:
4114:
4112:
4109:
4108:
4107:
4099:
4097:
4093:
4089:
4087:
4083:
4079:
4077:
4072:
4067:
4065:
4057:
4056:
4053:
4047:
4044:
4042:
4039:
4037:
4034:
4032:
4029:
4027:
4024:
4022:
4019:
4018:
4016:
4014:
4010:
4004:
4003:Orientability
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3980:
3978:
3974:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3931:
3928:
3926:
3923:
3922:
3921:
3918:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3890:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3870:
3867:
3866:
3864:
3860:
3855:
3845:
3842:
3840:
3839:Set-theoretic
3837:
3833:
3830:
3829:
3828:
3825:
3821:
3818:
3817:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3800:Combinatorial
3798:
3796:
3793:
3791:
3788:
3787:
3785:
3781:
3777:
3770:
3765:
3763:
3758:
3756:
3751:
3750:
3747:
3739:
3737:0-486-43479-6
3733:
3729:
3724:
3721:
3720:0-486-68735-X
3717:
3713:
3709:
3705:
3701:
3697:
3691:
3687:
3683:
3679:
3675:
3673:0-471-31716-0
3669:
3665:
3660:
3659:
3652:
3649:
3648:3-540-18178-4
3645:
3641:
3637:
3636:
3632:
3625:
3619:
3616:
3611:
3607:
3601:
3598:
3594:
3589:
3586:
3583:
3577:
3574:
3571:
3565:
3562:
3555:
3550:
3547:
3546:
3542:
3540:
3538:
3517:
3507:
3503:
3499:
3498:Cauchy spaces
3495:
3479:
3474:
3466:
3462:
3443:
3440:
3430:
3416:
3413:
3410:
3405:
3401:
3377:
3372:
3368:
3345:
3342:
3338:
3334:
3329:
3325:
3321:
3316:
3312:
3291:
3288:
3285:
3282:
3272:
3256:
3252:
3243:
3227:
3224:
3216:
3200:
3180:
3168:
3167:maximal ideal
3160:
3148:
3144:
3140:
3133:
3117:
3112:
3108:
3104:
3099:
3095:
3091:
3088:
3085:
3080:
3076:
3072:
3067:
3063:
3055:
3051:
3034:
3028:
3024:
3020:
3017:
3014:
3009:
3005:
3000:
2992:
2984:
2980:
2976:
2972:
2971:
2955:
2935:
2932:
2929:
2909:
2893:
2892:
2875:
2870:
2866:
2860:
2857:
2854:
2850:
2846:
2841:
2837:
2829:
2828:
2827:
2826:
2810:
2790:
2787:
2779:
2778:intersections
2775:
2759:
2734:
2731:
2728:
2725:
2722:
2715:
2711:
2706:
2702:
2693:
2677:
2652:
2649:
2646:
2643:
2640:
2633:
2629:
2624:
2620:
2599:
2577:
2573:
2565:of open sets
2564:
2546:
2542:
2521:
2501:
2497:
2496:
2492:
2488:
2472:
2467:
2464:
2461:
2457:
2453:
2448:
2444:
2440:
2435:
2431:
2408:
2404:
2381:
2377:
2368:
2356:
2339:
2331:
2324:
2300:
2289:
2274:
2254:
2229:
2222:
2201:
2198:
2178:
2153:
2146:
2138:the open set
2137:
2136:
2135:
2134:
2118:
2098:
2078:
2075:
2055:
2047:
2031:
2009:
2005:
1997:
1993:
1977:
1965:
1957:
1953:
1949:
1942:
1938:
1927:
1923:
1908:
1905:
1904:
1900:
1898:
1896:
1892:
1863:
1861:
1852:
1834:
1818:
1794:
1770:
1769:
1768:
1767:
1750:
1747:
1727:
1707:
1704:
1701:
1698:
1690:
1687:
1683:
1679:
1665:
1659:
1653:
1650:
1630:
1627:
1624:
1616:
1615:neighbourhood
1612:
1598:
1592:
1586:
1580:
1577:
1574:
1571:
1551:
1531:
1511:
1503:
1488:
1480:
1464:
1461:
1458:
1435:
1429:
1426:
1418:
1415:
1411:
1395:
1387:
1383:
1369:
1363:
1340:
1317:
1294:
1275:
1272:
1269:
1266:
1263:
1260:
1252:
1233:
1225:
1221:
1207:
1204:
1181:
1170:
1154:
1151:
1148:
1145:
1137:
1118:
1111:
1110:
1109:
1092:
1066:
1060:
1057:
1037:
1013:
1005:
1001:
997:
982:
974:
970:
956:
953:
933:
925:
924:neighbourhood
909:
901:
885:
866:
863:
860:
857:
837:
817:
809:
795:
792:
784:
769:
761:
745:
742:
739:
736:
728:
713:
705:
701:
698:
694:
679:
671:
657:
654:
646:
645:closed subset
627:
604:
601:
598:
595:
575:
567:
560:
540:
533:
514:
507:
506:
505:
491:
479:
477:
471:
463:
446:
444:
438:
436:
433:as a type of
432:
428:
420:
416:
411:Fréchet space
410:
408:
400:
396:
384:
380:
376:
369:is called an
368:
365:
362:
358:
350:
347:
346:
345:
343:
342:neighbourhood
338:
335:
333:
329:
325:
322:be points in
321:
317:
313:
309:
301:
299:
297:
285:
281:
273:
269:
265:
257:
253:
241:
238:
237:
234:
228:
223:
214:
208:
203:
194:
190:
185:
176:
172:
170:
159:
155:
150:
141:
137:
132:
123:
119:
117:
105:
101:
96:
87:
83:
78:
69:
65:
60:
51:
47:
43:
40:
35:
31:
19:
4133:Publications
3998:Chern number
3988:Betti number
3871: /
3862:Key concepts
3810:Differential
3727:
3711:
3685:
3657:
3639:
3633:Bibliography
3623:
3618:
3609:
3600:
3588:
3581:
3576:
3569:
3564:
3518:
3484:The terms "T
3483:
3215:prime ideals
3159:prime ideals
2367:intersection
1956:infinite set
1932:but is not T
1913:but is not T
1864:
1853:
1823:
1765:
1029:
483:
456:space and a
447:
422:
413:also has an
409:. (The term
406:
398:
394:
388:
382:
370:
366:
360:
352:
351:is called a
348:
331:
327:
323:
319:
315:
307:
305:
275:
272:neighborhood
259:
249:
126:completely T
71:
66:(Kolmogorov)
4096:Wikiversity
4013:Key results
3054:polynomials
2692:even number
1962:but is not
1686:closed sets
1617:of a point
995:is trivial.
900:limit point
437:. The term
377:if any two
302:Definitions
256:mathematics
173:(Tychonoff)
102:(Hausdorff)
4148:Categories
3942:CW complex
3883:Continuity
3873:Closed set
3832:cohomology
3523:, so the T
3467:space is T
1854:A finite T
1691:For every
1138:Given any
729:For every
699:is closed.
697:finite set
480:Properties
381:points in
46:Kolmogorov
4121:geometric
4116:algebraic
3967:Cobordism
3903:Hausdorff
3898:connected
3815:Geometric
3805:Continuum
3795:Algebraic
3704:175294365
3556:Citations
3516:spaces).
3381:∅
3322:∩
3286:∈
3132:Hausdorff
3105:−
3089:…
3073:−
3018:…
2981:(over an
2858:∈
2851:⋂
2726:−
2465:∪
2441:∩
2247:contains
2171:contains
1996:open sets
1964:Hausdorff
1893:is not a
1881:is also R
1805:⟹
1781:⟹
1771:separated
1702:∈
1654:
1643:contains
1628:∈
1596:∅
1581:
1575:∩
1479:partition
1462:∈
1430:
1419:The sets
1410:symmetric
1270:∈
1149:∈
1061:
861:∈
783:converges
740:∈
599:∈
441:also has
337:separated
120:(Urysohn)
84:(Fréchet)
4086:Wikibook
4064:Category
3952:Manifold
3920:Homotopy
3878:Interior
3869:Open set
3827:Homology
3776:Topology
3684:(1996).
3543:See also
3433:contain
3275:contain
3050:zero set
2489:but not
2267:and not
2191:but not
1992:integers
1901:Examples
1860:discrete
1819:distinct
1682:open set
1253:For any
785:only to
726:is open.
704:cofinite
561:and an R
401:and an R
314:and let
252:topology
18:T₁ space
4111:general
3913:uniform
3893:compact
3844:Digital
3213:of all
3052:of the
3048:is the
2563:subbase
2514:. Let
2487:compact
2048:subset
1564:, then
1477:form a
1169:closure
1131:is an R
706:set of
476:space.
462:) space
240:History
166:3½
4106:Topics
3908:metric
3783:Fields
3734:
3718:
3702:
3692:
3670:
3646:
3500:, and
3463:Every
3149:) is T
2985:) is T
2977:on an
2694:, and
2690:is an
2612:to be
2491:closed
2046:finite
1954:on an
1939:Every
1680:Every
1613:Every
1135:space.
702:Every
695:Every
565:space.
531:space.
527:is a T
472:is a T
225:
205:
187:
152:
134:
113:½
98:
80:
62:
3888:Space
3488:", "R
2948:(for
2774:basis
2510:nor R
1847:and T
898:is a
643:is a
559:space
553:is a
427:space
375:space
357:space
310:be a
290:and R
280:space
266:is a
264:space
3732:ISBN
3716:ISBN
3700:OCLC
3690:ISBN
3668:ISBN
3644:ISBN
3506:nets
3393:and
3360:and
3240:The
2973:The
2922:and
2396:and
2111:and
1950:The
1943:is T
1924:The
1451:for
1384:The
1246:is T
1222:The
1167:the
758:the
334:are
330:and
318:and
306:Let
258:, a
3664:116
3273:not
3217:of
2823:are
2752:if
2670:if
2423:is
2068:of
1504:If
1481:of
1408:is
1388:on
1226:of
1171:of
1030:If
975:to
926:of
902:of
830:of
762:at
647:of
484:If
445:.)
417:in
389:A T
250:In
37:in
4150::
3698:.
3666:.
3608:.
3539:.
3496:,
3431:do
3134:(T
2847::=
1966:(T
1851:.
1651:cl
1578:cl
1427:cl
1416:).
1108:)
1058:cl
298:.
3768:e
3761:t
3754:v
3740:.
3706:.
3676:.
3650:.
3612:.
3533:0
3529:0
3525:1
3521:0
3514:0
3510:1
3490:0
3486:1
3469:1
3458:1
3444:.
3441:a
3417:.
3414:X
3411:=
3406:1
3402:O
3378:=
3373:0
3369:O
3346:b
3343:a
3339:O
3335:=
3330:b
3326:O
3317:a
3313:O
3292:.
3289:A
3283:a
3257:a
3253:O
3228:.
3225:A
3201:X
3181:A
3171:1
3163:0
3155:1
3151:0
3136:2
3118:.
3113:n
3109:c
3100:n
3096:x
3092:,
3086:,
3081:1
3077:c
3068:1
3064:x
3035:)
3029:n
3025:c
3021:,
3015:,
3010:1
3006:c
3001:(
2987:1
2956:x
2936:1
2933:+
2930:x
2910:x
2900:1
2896:0
2876:.
2871:x
2867:G
2861:A
2855:x
2842:A
2838:U
2811:X
2791:,
2788:A
2760:x
2738:}
2735:x
2732:,
2729:1
2723:x
2720:{
2716:O
2712:=
2707:x
2703:G
2678:x
2656:}
2653:1
2650:+
2647:x
2644:,
2641:x
2638:{
2634:O
2630:=
2625:x
2621:G
2600:x
2578:x
2574:G
2547:A
2543:O
2522:X
2512:1
2508:1
2504:0
2473:,
2468:B
2462:A
2458:O
2454:=
2449:B
2445:O
2436:A
2432:O
2409:B
2405:O
2382:A
2378:O
2363:2
2359:1
2340:,
2335:}
2332:x
2329:{
2325:O
2304:}
2301:x
2298:{
2287:;
2275:y
2255:x
2233:}
2230:y
2227:{
2223:O
2202:,
2199:x
2179:y
2157:}
2154:x
2151:{
2147:O
2131::
2119:y
2099:x
2079:.
2076:X
2056:A
2032:X
2010:A
2006:O
1978:X
1968:2
1960:1
1945:1
1936:.
1934:1
1930:0
1921:.
1919:0
1915:1
1911:0
1887:2
1883:0
1879:0
1875:1
1871:1
1867:1
1856:1
1849:0
1845:0
1841:1
1837:1
1832:0
1830:T
1826:0
1751:.
1748:x
1728:x
1708:,
1705:X
1699:x
1688:.
1666:.
1663:}
1660:x
1657:{
1631:X
1625:x
1599:.
1593:=
1590:}
1587:x
1584:{
1572:F
1552:F
1532:x
1512:F
1489:X
1465:X
1459:x
1439:}
1436:x
1433:{
1396:X
1370:.
1367:}
1364:x
1361:{
1341:y
1321:}
1318:y
1315:{
1295:x
1276:,
1273:X
1267:y
1264:,
1261:x
1250:.
1248:1
1234:X
1208:.
1205:x
1185:}
1182:x
1179:{
1155:,
1152:X
1146:x
1133:0
1119:X
1096:}
1093:x
1090:{
1070:}
1067:x
1064:{
1038:X
1014:X
983:X
957:.
954:S
934:x
910:S
886:x
867:,
864:X
858:x
838:X
818:S
796:.
793:x
770:x
746:,
743:X
737:x
714:X
680:X
658:.
655:X
631:}
628:x
625:{
605:,
602:X
596:x
576:X
563:0
557:0
555:T
541:X
529:1
515:X
492:X
474:1
466:0
460:0
454:0
450:1
425:1
423:T
403:0
391:1
383:X
373:0
371:R
367:X
361:X
355:1
353:T
349:X
332:y
328:x
324:X
320:y
316:x
308:X
292:0
288:1
278:0
276:R
262:1
260:T
220:6
217:T
200:5
197:T
182:4
179:T
162:T
147:3
144:T
129:2
111:2
108:T
93:2
90:T
75:1
72:T
57:0
54:T
20:)
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