31:
175:
This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implications between increasingly subtle and complex variations on the notion of a locally connected space. As an example, the notion of local connectedness im kleinen at a point and its
4123:
An example of a space whose quasicomponents are not equal to its components is a sequence with a double limit point. This space is totally disconnected, but both limit points lie in the same quasicomponent, because any clopen set containing one of them must contain a tail of the sequence, and thus
2672:
has only finitely many connected components, then each component is the complement of a finite union of closed sets and therefore open. In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e.,
3039:
into pairwise disjoint open sets. It follows that an open connected subspace of a locally path connected space is necessarily path connected. Moreover, if a space is locally path connected, then it is also locally connected, so for all
4242:
918:
The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above. However, if a space is path connected im kleinen at each of its points, it is locally path connected.
3918:
3305:
766:, that is connected im kleinen at a particular point, but not locally connected at that point. However, if a space is connected im kleinen at each of its points, it is locally connected.
3998:
3823:
3576:
3522:
2938:
4110:
3356:
199:
is far more complex. From this modern perspective, the stronger property of local path connectedness turns out to be more important: for instance, in order for a space to admit a
4390:
4328:
2269:
2205:
1939:
1058:
959:
158:
2054:
3629:
2849:
2397:
1316:
4044:
3251:
3139:
3022:
2752:
2562:
2347:
4289:
3544:
3468:
3426:
3404:
3378:
3327:
1090:
448:
2710:
3174:
2311:
2164:
1972:
1172:
3852:
3067:
2967:
2463:
1029:
3777:
3735:
3691:
2887:
2809:
222:
space must be locally constant. In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general: for instance
4880:. For Hausdorff spaces, it is shown that any continuous function from a connected locally connected space into a connected space with a dispersion point is constant
4130:
3951:
3093:
2663:
2632:
2597:
2525:
2232:
2108:
2081:
1999:
1263:
1192:
1118:
575:
3212:
1832:
1790:
1747:
1645:
1602:
1499:
1456:
916:
760:
299:
3488:
3446:
1852:
1809:
1767:
1724:
1704:
1684:
1664:
1622:
1579:
1559:
1539:
1519:
1476:
1433:
1413:
1393:
1366:
1216:
893:
870:
850:
830:
809:
787:
737:
710:
690:
670:
649:
626:
604:
565:
545:
518:
497:
475:
415:
395:
368:
343:
321:
276:
256:
2016:
is locally connected if and only if it is discrete. This can be used to explain the aforementioned fact that the rational numbers are not locally connected.
4924:
2971:
However the closure of a path connected set need not be path connected: for instance, the topologist's sine curve is the closure of the open subset
965:
2987:, being homeomorphic to an interval on the real line, is certainly path connected. Moreover, the path components of the topologist's sine curve
4481:
Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (2016). "The
Mazurkiewicz distance and sets that are finitely connected at the boundary".
4822:
3861:
716:
if it is weakly locally connected at each of its points; as indicated below, this concept is in fact the same as being locally connected.
119:, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of
2716:) that are not discrete, like Cantor space. However, the connected components of a locally connected space are also open, and thus are
4853:
Coppin, C. A. (1972), "Continuous
Functions from a Connected Locally Connected Space into a Connected Space with a Dispersion Point",
4790:
4775:
4754:
111:
have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of
4406:
2565:
1369:
3267:
3578:
are the constant maps. In fact, any continuous map from a connected space to a totally disconnected space must be constant.
1128:
is connected and locally connected, but not path connected, and not path connected im kleinen at any point. It is in fact
4411:
4114:
Another class of spaces for which the quasicomponents agree with the components is the class of compact
Hausdorff spaces.
4053:
3221:
has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form
4809:
2013:
1064:
451:
347:
172:, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below).
4929:
1902:
1129:
2001:
is locally connected. In particular, since a single point is certainly locally connected, it follows that any
1891:. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
124:
3960:
3785:
3549:
3493:
2900:
4804:
3332:
179:
In the latter part of the twentieth century, research trends shifted to more intense study of spaces like
4340:
4247:
3406:
is connected, and the image of a connected space under a continuous map must be connected, the image of
3215:
4334:
2237:
2173:
1907:
1034:
935:
218:) are open. It follows, for instance, that a continuous function from a locally connected space to a
187:
to
Euclidean space) but have complicated global behavior. By this it is meant that although the basic
134:
3381:
2478:
2114:
2022:
2006:
371:
219:
184:
3854:
Overall we have the following containments among path components, components and quasicomponents at
3598:
2818:
2366:
1268:
4337:
is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed
4003:
3224:
3098:
2998:
2727:
2534:
2486:
2319:
70:
3527:
3451:
3409:
3387:
3361:
3310:
1073:
431:
4903:
4872:
4814:
4508:
4490:
2676:
196:
188:
2897:, so by the Lemma is itself path connected. Because path connected sets are connected, we have
3153:
3035:
are open. Therefore the path components of a locally path connected space give a partition of
2281:
2136:
2126:
The following result follows almost immediately from the definitions but will be quite useful:
1944:
1145:
4818:
4800:
4786:
4771:
4750:
4742:
4237:{\displaystyle (\{0\}\cup \{{\frac {1}{n}}:n\in \mathbb {Z} ^{+}\})\times \setminus \{(0,0)\}}
1107:
418:
215:
59:
3828:
3043:
2943:
2442:
1368:
is weakly locally connected. To show it is locally connected, it is enough to show that the
978:
4893:
4862:
4500:
3752:
3710:
3666:
2862:
2784:
1093:
116:
4832:
3929:
3071:
2641:
2610:
2575:
2503:
2210:
2086:
2059:
1977:
1221:
1177:
762:
The converse does not hold, as shown for example by a certain infinite union of decreasing
4828:
4046:
Since local path connectedness implies local connectedness, it follows that at all points
1895:
1140:
1137:
574:
Locally path connected spaces are locally connected. The converse does not hold (see the
428:
Local connectedness does not imply connectedness (consider two disjoint open intervals in
200:
169:
165:
112:
104:
3141:
That is, for a locally path connected space the components and path components coincide.
1096:
endowed with the standard
Euclidean topology, is neither connected nor locally connected.
3179:
1814:
1772:
1729:
1627:
1584:
1481:
1438:
898:
742:
281:
4763:
4416:
3473:
3431:
3218:
2002:
1864:
1837:
1794:
1752:
1709:
1689:
1669:
1649:
1607:
1564:
1544:
1524:
1504:
1461:
1418:
1398:
1378:
1351:
1201:
1195:
1121:
is connected and locally connected, but not path connected, nor locally path connected.
1111:
878:
855:
835:
815:
794:
772:
722:
695:
675:
655:
634:
611:
589:
550:
530:
521:
503:
482:
460:
400:
380:
353:
328:
306:
261:
241:
227:
92:
4898:
4867:
17:
4918:
4512:
108:
77:
30:
223:
1067:
is a subspace of the
Euclidean plane that is connected, but not locally connected.
1103:
is path connected but not locally path connected, and not even locally connected.
2717:
1125:
763:
55:
4504:
1100:
969:
192:
2754:
of its distinct connected components. Conversely, if for every open subset
1332:
A space is locally connected if and only if it is weakly locally connected.
450:
for example); and connectedness does not imply local connectedness (see the
176:
relation to local connectedness will be considered later on in the article.
115:, and the recognition of their independence from the particular form of the
4580:
4554:
712:
has a neighborhood base consisting of connected sets. A space is called
180:
74:
51:
42:
and it contains a connected open set (the dark green disk) that contains
961:
is locally path connected, thus locally connected; it is also connected.
4907:
4876:
164:> 1) proved to be much more complicated. Indeed, while any compact
3027:
A space is locally path connected if and only if for all open subsets
3738:
2770:
admits a base of connected sets and is therefore locally connected.
2415:
Evidently both relations are reflexive and symmetric. Moreover, if
2435:
are connected in a connected (respectively, path connected) subset
2423:
are contained in a connected (respectively, path connected) subset
567:
has a neighborhood base consisting of path connected open sets. A
4817:
reprint of 1978 ed.), Mineola, NY: Dover
Publications, Inc.,
4495:
4330:
are two different components which lie in the same quasicomponent.
29:
832:
contains a path connected (not necessarily open) neighborhood of
571:
is a space that is locally path connected at each of its points.
123:
subsets of
Euclidean space was understood quite early on via the
2465:
is a connected (respectively, path connected) subset containing
191:
of manifolds is relatively simple (as manifolds are essentially
206:
A space is locally connected if and only if for every open set
91:
if every point admits a neighbourhood basis consisting of open
4417:
It is conjectured that the
Mandelbrot set is locally connected
2005:
is locally connected. On the other hand, a discrete space is
3524:
Since this image is nonempty, the only continuous maps from '
672:
contains a connected (not necessarily open) neighborhood of
425:
is a space that is locally connected at each of its points.
4884:
2234:
is connected (respectively, path connected) then the union
968:
is locally connected, since each point has a local base of
872:
has a neighborhood base consisting of path connected sets.
1894:
A space is locally connected if and only if it admits a
4663:
Kelley, Theorem 20, p. 54; Willard, Theorem 26.8, p.193
3591:
be a topological space. We define a third relation on
3913:{\displaystyle PC_{x}\subseteq C_{x}\subseteq QC_{x}.}
3300:{\displaystyle f:\mathbb {R} \to \mathbb {R} _{\ell }}
3095:
is connected and open, hence path connected, that is,
2083:
is locally connected and all but finitely many of the
1887:
admits a neighborhood base of sets that have property
4343:
4292:
4250:
4133:
4056:
4006:
3963:
3932:
3864:
3831:
3788:
3755:
3737:
can also be characterized as the intersection of all
3713:
3669:
3601:
3552:
3530:
3496:
3476:
3454:
3434:
3412:
3390:
3364:
3335:
3313:
3270:
3227:
3182:
3156:
3101:
3074:
3046:
3001:
2946:
2903:
2865:
2821:
2787:
2730:
2679:
2644:
2613:
2578:
2537:
2506:
2445:
2369:
2322:
2284:
2240:
2213:
2176:
2139:
2089:
2062:
2025:
1980:
1947:
1910:
1840:
1817:
1797:
1775:
1755:
1732:
1712:
1692:
1672:
1652:
1630:
1610:
1587:
1567:
1547:
1527:
1507:
1484:
1464:
1441:
1421:
1401:
1381:
1354:
1271:
1224:
1204:
1180:
1148:
1076:
1037:
981:
938:
901:
881:
858:
838:
818:
797:
775:
745:
725:
698:
678:
658:
637:
614:
592:
553:
533:
506:
485:
463:
434:
403:
383:
356:
331:
309:
284:
264:
244:
195:
according to most definitions of the concept), their
137:
1867:
of topological spaces, i.e., a topological property
4650:
4648:
4646:
4644:
2889:is also the union of all path connected subsets of
2009:, so is connected only if it has at most one point.
1974:of spaces is locally connected if and only if each
4384:
4322:
4283:
4236:
4104:
4038:
3992:
3945:
3912:
3846:
3817:
3771:
3729:
3685:
3623:
3570:
3538:
3516:
3482:
3462:
3440:
3420:
3398:
3372:
3350:
3321:
3299:
3245:
3206:
3168:
3133:
3087:
3061:
3016:
2961:
2932:
2881:
2843:
2803:
2746:
2704:
2657:
2626:
2591:
2556:
2519:
2457:
2391:
2341:
2305:
2274:Now consider two relations on a topological space
2263:
2226:
2199:
2158:
2102:
2075:
2048:
1993:
1966:
1933:
1846:
1826:
1803:
1784:
1761:
1741:
1718:
1698:
1678:
1658:
1639:
1616:
1596:
1573:
1553:
1533:
1513:
1493:
1470:
1450:
1427:
1407:
1387:
1360:
1310:
1257:
1210:
1186:
1166:
1084:
1052:
1023:
953:
910:
887:
864:
844:
824:
803:
781:
754:
731:
704:
684:
664:
643:
620:
598:
559:
539:
512:
491:
469:
442:
409:
389:
362:
337:
315:
293:
270:
250:
152:
4698:
4696:
203:it must be connected and locally path connected.
4886:Proceedings of the American Mathematical Society
4855:Proceedings of the American Mathematical Society
4566:
4564:
4892:(5), American Mathematical Society: 1237–1241,
1119:lexicographic order topology on the unit square
576:lexicographic order topology on the unit square
4437:
4435:
4433:
4431:
4244:is locally compact and Hausdorff but the sets
4861:(2), American Mathematical Society: 625–626,
2720:. It follows that a locally connected space
2489:. We consider these two partitions in turn.
2271:is connected (respectively, path connected).
8:
4379:
4373:
4299:
4293:
4257:
4251:
4231:
4213:
4183:
4149:
4143:
4137:
3234:
3228:
2699:
2693:
2153:
2140:
1961:
1948:
1060:is locally path connected but not connected.
4555:"Show that X is not locally connected at p"
3779:is closed; in general it need not be open.
3448:must be connected. Therefore, the image of
1326:
183:, which are locally well understood (being
4581:"Definition of locally pathwise connected"
4535:Steen & Seebach, example 119.4, p. 139
4050:of a locally path connected space we have
2599:is the unique maximal connected subset of
1541:so that there is a connected neighborhood
1336:
1106:A countably infinite set endowed with the
875:A space that is locally path connected at
4897:
4866:
4494:
4364:
4351:
4342:
4291:
4249:
4177:
4173:
4172:
4152:
4132:
4093:
4077:
4064:
4055:
4027:
4014:
4005:
3984:
3971:
3962:
3937:
3931:
3901:
3885:
3872:
3863:
3830:
3809:
3793:
3787:
3763:
3754:
3721:
3712:
3677:
3668:
3609:
3600:
3559:
3555:
3554:
3551:
3532:
3531:
3529:
3509:
3503:
3499:
3498:
3495:
3475:
3456:
3455:
3453:
3433:
3414:
3413:
3411:
3392:
3391:
3389:
3366:
3365:
3363:
3342:
3338:
3337:
3334:
3315:
3314:
3312:
3291:
3287:
3286:
3278:
3277:
3269:
3226:
3181:
3155:
3122:
3106:
3100:
3079:
3073:
3045:
3000:
2945:
2924:
2911:
2902:
2873:
2864:
2829:
2820:
2795:
2786:
2738:
2729:
2684:
2678:
2649:
2643:
2618:
2612:
2583:
2577:
2545:
2536:
2511:
2505:
2444:
2377:
2368:
2330:
2321:
2283:
2255:
2245:
2239:
2218:
2212:
2191:
2181:
2175:
2147:
2138:
2094:
2088:
2067:
2061:
2056:is locally connected if and only if each
2040:
2030:
2024:
1985:
1979:
1955:
1946:
1925:
1915:
1909:
1863:Local connectedness is, by definition, a
1839:
1816:
1796:
1774:
1754:
1731:
1711:
1691:
1671:
1651:
1629:
1609:
1586:
1566:
1546:
1526:
1506:
1483:
1463:
1440:
1420:
1400:
1380:
1353:
1270:
1223:
1203:
1179:
1174:is locally path-connected if and only if
1147:
1078:
1077:
1075:
1044:
1040:
1039:
1036:
980:
945:
941:
940:
937:
900:
880:
857:
837:
817:
796:
774:
744:
724:
697:
677:
657:
636:
613:
591:
552:
532:
505:
484:
462:
436:
435:
433:
402:
382:
355:
330:
308:
283:
263:
243:
144:
140:
139:
136:
4427:
4210:
3659:. This is an equivalence relation on
3005:
2399:if there is a path connected subset of
1339:
966:locally convex topological vector space
4611:Steen & Seebach, example 48, p. 73
3926:is locally connected, then, as above,
2634:is also a connected subset containing
1348:For the non-trivial direction, assume
421:consisting of connected open sets. A
4476:
4474:
3993:{\displaystyle QC_{x}\subseteq C_{x}}
3818:{\displaystyle C_{x}\subseteq QC_{x}}
3571:{\displaystyle \mathbb {R} _{\ell },}
3517:{\displaystyle \mathbb {R} _{\ell }/}
2933:{\displaystyle PC_{x}\subseteq C_{x}}
719:A space that is locally connected at
7:
4105:{\displaystyle PC_{x}=C_{x}=QC_{x}.}
3351:{\displaystyle \mathbb {R} _{\ell }}
2995:, which is open but not closed, and
2117:is locally connected, and connected.
1686:(the connected component containing
972:(and hence connected) neighborhoods.
103:Throughout the history of topology,
3490:must be a subset of a component of
4385:{\displaystyle QC_{x}=C_{x}=\{x\}}
4284:{\displaystyle \{0\}\times [-1,0)}
2349:if there is a connected subset of
25:
4899:10.1090/s0002-9939-1968-0254814-3
4868:10.1090/S0002-9939-1972-0296913-7
4720:Engelking, Theorem 6.1.23, p. 357
4654:Willard, Problem 26B, pp. 195–196
4468:Willard, Definition 27.14, p. 201
4323:{\displaystyle \{0\}\times (0,1]}
2264:{\displaystyle \bigcup _{i}Y_{i}}
2200:{\displaystyle \bigcap _{i}Y_{i}}
1934:{\displaystyle \coprod _{i}X_{i}}
1114:) but not locally path connected.
4925:Properties of topological spaces
4681:Willard, Corollary 27.10, p. 200
4638:Willard, Definition 26.11, p.194
4593:Steen & Seebach, pp. 137–138
4450:Willard, Definition 27.7, p. 199
2724:is a topological disjoint union
1053:{\displaystyle \mathbb {R} ^{1}}
954:{\displaystyle \mathbb {R} ^{n}}
895:is path connected im kleinen at
258:be a topological space, and let
226:is totally disconnected but not
153:{\displaystyle \mathbb {R} ^{n}}
83:As a stronger notion, the space
4785:(2nd ed.), Prentice Hall,
4459:Willard, Definition 27.4, p.199
2049:{\displaystyle \prod _{i}X_{i}}
4729:Steen & Seebach, pp. 54-55
4672:Willard, Theorem 26.12, p. 194
4629:Willard, Theorem 26.7a, p. 192
4620:Willard, theorem 27.13, p. 201
4602:Steen & Seebach, pp. 49–50
4570:Willard, Theorem 27.16, p. 201
4407:Locally simply connected space
4317:
4305:
4278:
4263:
4228:
4216:
4207:
4192:
4186:
4134:
3624:{\displaystyle x\equiv _{qc}y}
3282:
3201:
3189:
3024:which is closed but not open.
2844:{\displaystyle y\equiv _{pc}x}
2762:, the connected components of
2439:, then the Lemma implies that
2392:{\displaystyle x\equiv _{pc}y}
2122:Components and path components
1311:{\displaystyle \to (X,\tau ).}
1302:
1290:
1287:
1284:
1272:
1252:
1243:
1231:
1228:
1161:
1149:
1110:is locally connected (indeed,
1018:
1006:
1000:
988:
210:, the connected components of
27:Property of topological spaces
1:
4749:. Heldermann Verlag, Berlin.
4711:Willard, Theorem 27.5, p. 199
4690:Willard, Theorem 27.9, p. 200
4483:Journal of Geometric Analysis
4412:Semi-locally simply connected
4039:{\displaystyle QC_{x}=C_{x}.}
3631:if there is no separation of
3253:is a path component for each
3246:{\displaystyle \{a\}\times I}
3134:{\displaystyle C_{x}=PC_{x}.}
3017:{\displaystyle C\setminus U,}
2747:{\displaystyle \coprod C_{x}}
2557:{\displaystyle y\equiv _{c}x}
2481:, and defines a partition of
2342:{\displaystyle x\equiv _{c}y}
791:path connected im kleinen at
4702:Willard, Problem 27D, p. 202
3539:{\displaystyle \mathbb {R} }
3463:{\displaystyle \mathbb {R} }
3421:{\displaystyle \mathbb {R} }
3399:{\displaystyle \mathbb {R} }
3373:{\displaystyle \mathbb {R} }
3322:{\displaystyle \mathbb {R} }
2477:. Thus each relation is an
2207:is nonempty. Then, if each
1898:of (open) connected subsets.
1435:be a connected component of
1085:{\displaystyle \mathbb {Q} }
631:weakly locally connected at
569:locally path connected space
443:{\displaystyle \mathbb {R} }
4842:; Dover Publications, 2004.
4810:Counterexamples in Topology
4544:Munkres, exercise 7, p. 162
4526:Munkres, exercise 6, p. 162
3953:is a clopen set containing
2705:{\displaystyle C_{x}=\{x\}}
739:is connected im kleinen at
34:In this topological space,
4946:
3663:and the equivalence class
2572:. The Lemma implies that
2014:totally disconnected space
1879:if and only if each point
1769:was an arbitrary point of
1624:is connected and contains
479:locally path connected at
4505:10.1007/s12220-015-9575-9
3307:be a continuous map from
3169:{\displaystyle I\times I}
3031:, the path components of
2975:consisting of all points
2306:{\displaystyle x,y\in X,}
2159:{\displaystyle \{Y_{i}\}}
2019:A nonempty product space
1967:{\displaystyle \{X_{i}\}}
1167:{\displaystyle (X,\tau )}
1130:totally path disconnected
928:For any positive integer
812:if every neighborhood of
652:if every neighborhood of
500:if every neighborhood of
2607:. Since the closure of
1726:is an interior point of
1265:of all continuous paths
852:, that is, if the point
714:weakly locally connected
692:, that is, if the point
608:connected im kleinen at
582:Connectedness im kleinen
547:, that is, if the point
397:, that is, if the point
69:if every point admits a
4781:Munkres, James (1999),
3847:{\displaystyle x\in X.}
3062:{\displaystyle x\in X,}
2962:{\displaystyle x\in X.}
2458:{\displaystyle A\cup B}
2166:a family of subsets of
1372:of open sets are open.
1065:topologist's sine curve
1024:{\displaystyle S=\cup }
452:topologist's sine curve
423:locally connected space
4805:Seebach, J. Arthur Jr.
4386:
4324:
4285:
4238:
4106:
4040:
3994:
3947:
3914:
3848:
3819:
3773:
3772:{\displaystyle QC_{x}}
3731:
3730:{\displaystyle QC_{x}}
3687:
3686:{\displaystyle QC_{x}}
3625:
3572:
3540:
3518:
3484:
3464:
3442:
3422:
3400:
3374:
3352:
3323:
3301:
3247:
3208:
3170:
3135:
3089:
3063:
3018:
2963:
2934:
2883:
2882:{\displaystyle PC_{x}}
2845:
2805:
2804:{\displaystyle PC_{x}}
2748:
2706:
2659:
2628:
2593:
2558:
2521:
2459:
2393:
2343:
2307:
2265:
2228:
2201:
2160:
2104:
2077:
2050:
1995:
1968:
1935:
1854:is locally connected.
1848:
1828:
1805:
1786:
1763:
1743:
1720:
1700:
1680:
1660:
1641:
1618:
1598:
1575:
1555:
1535:
1515:
1495:
1472:
1452:
1429:
1409:
1389:
1362:
1312:
1259:
1212:
1188:
1168:
1086:
1054:
1025:
964:More generally, every
955:
932:, the Euclidean space
912:
889:
866:
846:
826:
805:
783:
756:
733:
706:
686:
666:
645:
622:
600:
561:
541:
514:
493:
471:
444:
411:
391:
364:
339:
317:
295:
272:
252:
154:
89:locally path connected
54:and other branches of
47:
38:is a neighbourhood of
18:Connected space/Proofs
4387:
4325:
4286:
4239:
4107:
4041:
3995:
3948:
3946:{\displaystyle C_{x}}
3915:
3849:
3820:
3774:
3732:
3688:
3626:
3573:
3541:
3519:
3485:
3465:
3443:
3423:
3401:
3375:
3353:
3324:
3302:
3248:
3209:
3171:
3136:
3090:
3088:{\displaystyle C_{x}}
3064:
3019:
2964:
2935:
2884:
2846:
2806:
2749:
2707:
2660:
2658:{\displaystyle C_{x}}
2629:
2627:{\displaystyle C_{x}}
2594:
2592:{\displaystyle C_{x}}
2559:
2522:
2520:{\displaystyle C_{x}}
2460:
2394:
2344:
2308:
2266:
2229:
2227:{\displaystyle Y_{i}}
2202:
2161:
2105:
2103:{\displaystyle X_{i}}
2078:
2076:{\displaystyle X_{i}}
2051:
1996:
1994:{\displaystyle X_{i}}
1969:
1936:
1849:
1829:
1806:
1787:
1764:
1744:
1721:
1701:
1681:
1661:
1642:
1619:
1599:
1576:
1556:
1536:
1521:is a neighborhood of
1516:
1496:
1473:
1453:
1430:
1410:
1390:
1363:
1313:
1260:
1258:{\displaystyle C(;X)}
1213:
1189:
1187:{\displaystyle \tau }
1169:
1087:
1055:
1026:
956:
913:
890:
867:
847:
827:
806:
784:
757:
734:
707:
687:
667:
646:
623:
601:
562:
542:
515:
494:
472:
445:
412:
392:
365:
340:
325:locally connected at
318:
296:
273:
253:
155:
33:
4341:
4290:
4248:
4131:
4124:the other point too.
4054:
4004:
3961:
3930:
3862:
3829:
3786:
3753:
3711:
3667:
3599:
3550:
3528:
3494:
3474:
3452:
3432:
3410:
3388:
3382:lower limit topology
3362:
3333:
3311:
3268:
3225:
3180:
3154:
3099:
3072:
3044:
2999:
2944:
2901:
2863:
2819:
2785:
2728:
2677:
2642:
2611:
2576:
2535:
2504:
2479:equivalence relation
2443:
2367:
2320:
2282:
2238:
2211:
2174:
2137:
2115:hyperconnected space
2087:
2060:
2023:
2007:totally disconnected
1978:
1945:
1908:
1838:
1815:
1795:
1773:
1753:
1730:
1710:
1690:
1670:
1666:must be a subset of
1650:
1628:
1608:
1585:
1565:
1545:
1525:
1505:
1482:
1462:
1439:
1419:
1399:
1379:
1370:connected components
1352:
1269:
1222:
1202:
1178:
1146:
1074:
1035:
979:
936:
899:
879:
856:
836:
816:
795:
773:
743:
723:
696:
676:
656:
635:
612:
590:
551:
531:
504:
483:
461:
432:
401:
381:
354:
329:
307:
282:
262:
242:
220:totally disconnected
185:locally homeomorphic
135:
2566:connected component
2487:equivalence classes
1875:possesses property
1330: —
1218:induced by the set
125:Heine–Borel theorem
71:neighbourhood basis
4801:Steen, Lynn Arthur
4743:Engelking, Ryszard
4382:
4320:
4281:
4234:
4102:
4036:
3990:
3943:
3910:
3844:
3815:
3769:
3727:
3683:
3621:
3568:
3536:
3514:
3480:
3460:
3438:
3418:
3396:
3370:
3348:
3319:
3297:
3243:
3207:{\displaystyle I=}
3204:
3166:
3131:
3085:
3059:
3014:
2959:
2930:
2879:
2841:
2801:
2744:
2702:
2655:
2638:, it follows that
2624:
2589:
2554:
2517:
2455:
2389:
2339:
2303:
2261:
2250:
2224:
2197:
2186:
2156:
2100:
2073:
2046:
2035:
1991:
1964:
1931:
1920:
1871:such that a space
1844:
1827:{\displaystyle X.}
1824:
1801:
1785:{\displaystyle C,}
1782:
1759:
1742:{\displaystyle C.}
1739:
1716:
1696:
1676:
1656:
1640:{\displaystyle x,}
1637:
1614:
1597:{\displaystyle U.}
1594:
1571:
1551:
1531:
1511:
1494:{\displaystyle C.}
1491:
1468:
1451:{\displaystyle U.}
1448:
1425:
1405:
1385:
1358:
1328:
1308:
1255:
1208:
1184:
1164:
1082:
1050:
1021:
951:
911:{\displaystyle x.}
908:
885:
862:
842:
822:
801:
779:
755:{\displaystyle x.}
752:
729:
702:
682:
662:
641:
618:
596:
557:
537:
510:
489:
467:
440:
407:
387:
360:
335:
313:
294:{\displaystyle X.}
291:
268:
248:
197:algebraic topology
189:point-set topology
150:
48:
4838:Stephen Willard;
4824:978-0-486-68735-3
4160:
3655:is an element of
3647:is an element of
3483:{\displaystyle f}
3441:{\displaystyle f}
2241:
2177:
2026:
1911:
1859:
1858:
1847:{\displaystyle X}
1804:{\displaystyle C}
1762:{\displaystyle x}
1719:{\displaystyle x}
1699:{\displaystyle x}
1679:{\displaystyle C}
1659:{\displaystyle V}
1617:{\displaystyle V}
1574:{\displaystyle x}
1554:{\displaystyle V}
1534:{\displaystyle x}
1514:{\displaystyle U}
1478:be an element of
1471:{\displaystyle x}
1428:{\displaystyle C}
1408:{\displaystyle X}
1388:{\displaystyle U}
1361:{\displaystyle X}
1211:{\displaystyle X}
1108:cofinite topology
1031:of the real line
888:{\displaystyle x}
865:{\displaystyle x}
845:{\displaystyle x}
825:{\displaystyle x}
804:{\displaystyle x}
782:{\displaystyle X}
732:{\displaystyle x}
705:{\displaystyle x}
685:{\displaystyle x}
665:{\displaystyle x}
644:{\displaystyle x}
621:{\displaystyle x}
599:{\displaystyle X}
560:{\displaystyle x}
540:{\displaystyle x}
513:{\displaystyle x}
492:{\displaystyle x}
470:{\displaystyle X}
419:neighborhood base
410:{\displaystyle x}
390:{\displaystyle x}
363:{\displaystyle x}
338:{\displaystyle x}
316:{\displaystyle X}
271:{\displaystyle x}
251:{\displaystyle X}
216:subspace topology
67:locally connected
60:topological space
16:(Redirected from
4937:
4930:General topology
4910:
4901:
4879:
4870:
4841:
4840:General Topology
4835:
4795:
4769:
4768:General Topology
4760:
4747:General Topology
4730:
4727:
4721:
4718:
4712:
4709:
4703:
4700:
4691:
4688:
4682:
4679:
4673:
4670:
4664:
4661:
4655:
4652:
4639:
4636:
4630:
4627:
4621:
4618:
4612:
4609:
4603:
4600:
4594:
4591:
4585:
4584:
4577:
4571:
4568:
4559:
4558:
4551:
4545:
4542:
4536:
4533:
4527:
4524:
4518:
4516:
4498:
4478:
4469:
4466:
4460:
4457:
4451:
4448:
4442:
4439:
4391:
4389:
4388:
4383:
4369:
4368:
4356:
4355:
4335:Arens–Fort space
4329:
4327:
4326:
4321:
4288:
4287:
4282:
4243:
4241:
4240:
4235:
4182:
4181:
4176:
4161:
4153:
4111:
4109:
4108:
4103:
4098:
4097:
4082:
4081:
4069:
4068:
4045:
4043:
4042:
4037:
4032:
4031:
4019:
4018:
3999:
3997:
3996:
3991:
3989:
3988:
3976:
3975:
3952:
3950:
3949:
3944:
3942:
3941:
3919:
3917:
3916:
3911:
3906:
3905:
3890:
3889:
3877:
3876:
3853:
3851:
3850:
3845:
3824:
3822:
3821:
3816:
3814:
3813:
3798:
3797:
3778:
3776:
3775:
3770:
3768:
3767:
3736:
3734:
3733:
3728:
3726:
3725:
3692:
3690:
3689:
3684:
3682:
3681:
3630:
3628:
3627:
3622:
3617:
3616:
3577:
3575:
3574:
3569:
3564:
3563:
3558:
3545:
3543:
3542:
3537:
3535:
3523:
3521:
3520:
3515:
3513:
3508:
3507:
3502:
3489:
3487:
3486:
3481:
3469:
3467:
3466:
3461:
3459:
3447:
3445:
3444:
3439:
3427:
3425:
3424:
3419:
3417:
3405:
3403:
3402:
3397:
3395:
3379:
3377:
3376:
3371:
3369:
3357:
3355:
3354:
3349:
3347:
3346:
3341:
3328:
3326:
3325:
3320:
3318:
3306:
3304:
3303:
3298:
3296:
3295:
3290:
3281:
3252:
3250:
3249:
3244:
3213:
3211:
3210:
3205:
3175:
3173:
3172:
3167:
3140:
3138:
3137:
3132:
3127:
3126:
3111:
3110:
3094:
3092:
3091:
3086:
3084:
3083:
3068:
3066:
3065:
3060:
3023:
3021:
3020:
3015:
2968:
2966:
2965:
2960:
2939:
2937:
2936:
2931:
2929:
2928:
2916:
2915:
2888:
2886:
2885:
2880:
2878:
2877:
2850:
2848:
2847:
2842:
2837:
2836:
2810:
2808:
2807:
2802:
2800:
2799:
2753:
2751:
2750:
2745:
2743:
2742:
2711:
2709:
2708:
2703:
2689:
2688:
2664:
2662:
2661:
2656:
2654:
2653:
2633:
2631:
2630:
2625:
2623:
2622:
2598:
2596:
2595:
2590:
2588:
2587:
2563:
2561:
2560:
2555:
2550:
2549:
2526:
2524:
2523:
2518:
2516:
2515:
2464:
2462:
2461:
2456:
2403:containing both
2398:
2396:
2395:
2390:
2385:
2384:
2353:containing both
2348:
2346:
2345:
2340:
2335:
2334:
2312:
2310:
2309:
2304:
2270:
2268:
2267:
2262:
2260:
2259:
2249:
2233:
2231:
2230:
2225:
2223:
2222:
2206:
2204:
2203:
2198:
2196:
2195:
2185:
2170:. Suppose that
2165:
2163:
2162:
2157:
2152:
2151:
2133:be a space, and
2109:
2107:
2106:
2101:
2099:
2098:
2082:
2080:
2079:
2074:
2072:
2071:
2055:
2053:
2052:
2047:
2045:
2044:
2034:
2000:
1998:
1997:
1992:
1990:
1989:
1973:
1971:
1970:
1965:
1960:
1959:
1940:
1938:
1937:
1932:
1930:
1929:
1919:
1853:
1851:
1850:
1845:
1833:
1831:
1830:
1825:
1810:
1808:
1807:
1802:
1791:
1789:
1788:
1783:
1768:
1766:
1765:
1760:
1748:
1746:
1745:
1740:
1725:
1723:
1722:
1717:
1705:
1703:
1702:
1697:
1685:
1683:
1682:
1677:
1665:
1663:
1662:
1657:
1646:
1644:
1643:
1638:
1623:
1621:
1620:
1615:
1603:
1601:
1600:
1595:
1580:
1578:
1577:
1572:
1560:
1558:
1557:
1552:
1540:
1538:
1537:
1532:
1520:
1518:
1517:
1512:
1500:
1498:
1497:
1492:
1477:
1475:
1474:
1469:
1457:
1455:
1454:
1449:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1367:
1365:
1364:
1359:
1337:
1331:
1317:
1315:
1314:
1309:
1264:
1262:
1261:
1256:
1217:
1215:
1214:
1209:
1194:is equal to the
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1094:rational numbers
1091:
1089:
1088:
1083:
1081:
1059:
1057:
1056:
1051:
1049:
1048:
1043:
1030:
1028:
1027:
1022:
960:
958:
957:
952:
950:
949:
944:
917:
915:
914:
909:
894:
892:
891:
886:
871:
869:
868:
863:
851:
849:
848:
843:
831:
829:
828:
823:
810:
808:
807:
802:
788:
786:
785:
780:
761:
759:
758:
753:
738:
736:
735:
730:
711:
709:
708:
703:
691:
689:
688:
683:
671:
669:
668:
663:
650:
648:
647:
642:
627:
625:
624:
619:
605:
603:
602:
597:
566:
564:
563:
558:
546:
544:
543:
538:
527:neighborhood of
519:
517:
516:
511:
498:
496:
495:
490:
476:
474:
473:
468:
449:
447:
446:
441:
439:
416:
414:
413:
408:
396:
394:
393:
388:
377:neighborhood of
369:
367:
366:
361:
344:
342:
341:
336:
322:
320:
319:
314:
300:
298:
297:
292:
277:
275:
274:
269:
257:
255:
254:
249:
213:
209:
159:
157:
156:
151:
149:
148:
143:
117:Euclidean metric
21:
4945:
4944:
4940:
4939:
4938:
4936:
4935:
4934:
4915:
4914:
4883:
4852:
4849:
4847:Further reading
4839:
4825:
4799:
4793:
4780:
4767:
4757:
4741:
4738:
4733:
4728:
4724:
4719:
4715:
4710:
4706:
4701:
4694:
4689:
4685:
4680:
4676:
4671:
4667:
4662:
4658:
4653:
4642:
4637:
4633:
4628:
4624:
4619:
4615:
4610:
4606:
4601:
4597:
4592:
4588:
4579:
4578:
4574:
4569:
4562:
4553:
4552:
4548:
4543:
4539:
4534:
4530:
4525:
4521:
4480:
4479:
4472:
4467:
4463:
4458:
4454:
4449:
4445:
4441:Munkres, p. 161
4440:
4429:
4425:
4403:
4392:for all points
4360:
4347:
4339:
4338:
4246:
4245:
4171:
4129:
4128:
4120:
4089:
4073:
4060:
4052:
4051:
4023:
4010:
4002:
4001:
3980:
3967:
3959:
3958:
3933:
3928:
3927:
3897:
3881:
3868:
3860:
3859:
3827:
3826:
3805:
3789:
3784:
3783:
3759:
3751:
3750:
3717:
3709:
3708:
3673:
3665:
3664:
3635:into open sets
3605:
3597:
3596:
3585:
3583:Quasicomponents
3553:
3548:
3547:
3526:
3525:
3497:
3492:
3491:
3472:
3471:
3450:
3449:
3430:
3429:
3408:
3407:
3386:
3385:
3360:
3359:
3336:
3331:
3330:
3309:
3308:
3285:
3266:
3265:
3223:
3222:
3178:
3177:
3152:
3151:
3147:
3118:
3102:
3097:
3096:
3075:
3070:
3069:
3042:
3041:
2997:
2996:
2942:
2941:
2920:
2907:
2899:
2898:
2869:
2861:
2860:
2825:
2817:
2816:
2791:
2783:
2782:
2766:are open, then
2734:
2726:
2725:
2712:for all points
2680:
2675:
2674:
2645:
2640:
2639:
2614:
2609:
2608:
2579:
2574:
2573:
2541:
2533:
2532:
2507:
2502:
2501:
2441:
2440:
2373:
2365:
2364:
2326:
2318:
2317:
2280:
2279:
2251:
2236:
2235:
2214:
2209:
2208:
2187:
2172:
2171:
2143:
2135:
2134:
2124:
2090:
2085:
2084:
2063:
2058:
2057:
2036:
2021:
2020:
1981:
1976:
1975:
1951:
1943:
1942:
1921:
1906:
1905:
1860:
1836:
1835:
1813:
1812:
1793:
1792:
1771:
1770:
1751:
1750:
1728:
1727:
1708:
1707:
1688:
1687:
1668:
1667:
1648:
1647:
1626:
1625:
1606:
1605:
1583:
1582:
1563:
1562:
1543:
1542:
1523:
1522:
1503:
1502:
1480:
1479:
1460:
1459:
1437:
1436:
1417:
1416:
1397:
1396:
1377:
1376:
1350:
1349:
1342:
1334:
1329:
1323:
1267:
1266:
1220:
1219:
1200:
1199:
1176:
1175:
1144:
1143:
1141:Hausdorff space
1138:first-countable
1072:
1071:
1038:
1033:
1032:
977:
976:
939:
934:
933:
925:
897:
896:
877:
876:
854:
853:
834:
833:
814:
813:
793:
792:
771:
770:
741:
740:
721:
720:
694:
693:
674:
673:
654:
653:
633:
632:
610:
609:
588:
587:
584:
549:
548:
529:
528:
502:
501:
481:
480:
459:
458:
430:
429:
399:
398:
379:
378:
352:
351:
327:
326:
305:
304:
280:
279:
260:
259:
240:
239:
236:
211:
207:
201:universal cover
170:locally compact
166:Hausdorff space
138:
133:
132:
113:Euclidean space
101:
28:
23:
22:
15:
12:
11:
5:
4943:
4941:
4933:
4932:
4927:
4917:
4916:
4913:
4912:
4881:
4848:
4845:
4844:
4843:
4836:
4823:
4797:
4791:
4778:
4764:John L. Kelley
4761:
4755:
4737:
4734:
4732:
4731:
4722:
4713:
4704:
4692:
4683:
4674:
4665:
4656:
4640:
4631:
4622:
4613:
4604:
4595:
4586:
4572:
4560:
4546:
4537:
4528:
4519:
4489:(2): 873–897.
4470:
4461:
4452:
4443:
4426:
4424:
4421:
4420:
4419:
4414:
4409:
4402:
4399:
4398:
4397:
4381:
4378:
4375:
4372:
4367:
4363:
4359:
4354:
4350:
4346:
4331:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4180:
4175:
4170:
4167:
4164:
4159:
4156:
4151:
4148:
4145:
4142:
4139:
4136:
4125:
4119:
4116:
4101:
4096:
4092:
4088:
4085:
4080:
4076:
4072:
4067:
4063:
4059:
4035:
4030:
4026:
4022:
4017:
4013:
4009:
3987:
3983:
3979:
3974:
3970:
3966:
3940:
3936:
3909:
3904:
3900:
3896:
3893:
3888:
3884:
3880:
3875:
3871:
3867:
3843:
3840:
3837:
3834:
3812:
3808:
3804:
3801:
3796:
3792:
3766:
3762:
3758:
3749:. Accordingly
3724:
3720:
3716:
3699:quasicomponent
3697:is called the
3680:
3676:
3672:
3620:
3615:
3612:
3608:
3604:
3584:
3581:
3580:
3579:
3567:
3562:
3557:
3534:
3512:
3506:
3501:
3479:
3458:
3437:
3416:
3394:
3368:
3345:
3340:
3317:
3294:
3289:
3284:
3280:
3276:
3273:
3262:
3242:
3239:
3236:
3233:
3230:
3219:order topology
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3165:
3162:
3159:
3146:
3143:
3130:
3125:
3121:
3117:
3114:
3109:
3105:
3082:
3078:
3058:
3055:
3052:
3049:
3013:
3010:
3007:
3004:
2958:
2955:
2952:
2949:
2927:
2923:
2919:
2914:
2910:
2906:
2876:
2872:
2868:
2853:path component
2851:is called the
2840:
2835:
2832:
2828:
2824:
2811:of all points
2798:
2794:
2790:
2741:
2737:
2733:
2701:
2698:
2695:
2692:
2687:
2683:
2652:
2648:
2621:
2617:
2586:
2582:
2564:is called the
2553:
2548:
2544:
2540:
2527:of all points
2514:
2510:
2454:
2451:
2448:
2413:
2412:
2388:
2383:
2380:
2376:
2372:
2362:
2338:
2333:
2329:
2325:
2302:
2299:
2296:
2293:
2290:
2287:
2258:
2254:
2248:
2244:
2221:
2217:
2194:
2190:
2184:
2180:
2155:
2150:
2146:
2142:
2123:
2120:
2119:
2118:
2111:
2110:are connected.
2097:
2093:
2070:
2066:
2043:
2039:
2033:
2029:
2017:
2012:Conversely, a
2010:
2003:discrete space
1988:
1984:
1963:
1958:
1954:
1950:
1928:
1924:
1918:
1914:
1903:disjoint union
1899:
1892:
1865:local property
1857:
1856:
1843:
1823:
1820:
1800:
1781:
1778:
1758:
1738:
1735:
1715:
1695:
1675:
1655:
1636:
1633:
1613:
1593:
1590:
1570:
1550:
1530:
1510:
1490:
1487:
1467:
1447:
1444:
1424:
1404:
1384:
1357:
1344:
1343:
1340:
1335:
1324:
1322:
1319:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1207:
1196:final topology
1183:
1163:
1160:
1157:
1154:
1151:
1134:
1133:
1122:
1115:
1112:hyperconnected
1104:
1097:
1080:
1068:
1061:
1047:
1042:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
973:
962:
948:
943:
924:
923:First examples
921:
907:
904:
884:
861:
841:
821:
800:
789:is said to be
778:
751:
748:
728:
701:
681:
661:
640:
617:
595:
583:
580:
556:
536:
522:path connected
509:
488:
466:
438:
406:
386:
359:
334:
312:
290:
287:
278:be a point of
267:
247:
235:
232:
147:
142:
100:
97:
93:path connected
73:consisting of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4942:
4931:
4928:
4926:
4923:
4922:
4920:
4909:
4905:
4900:
4895:
4891:
4887:
4882:
4878:
4874:
4869:
4864:
4860:
4856:
4851:
4850:
4846:
4837:
4834:
4830:
4826:
4820:
4816:
4812:
4811:
4806:
4802:
4798:
4794:
4792:0-13-181629-2
4788:
4784:
4779:
4777:
4776:0-387-90125-6
4773:
4765:
4762:
4758:
4756:3-88538-006-4
4752:
4748:
4744:
4740:
4739:
4735:
4726:
4723:
4717:
4714:
4708:
4705:
4699:
4697:
4693:
4687:
4684:
4678:
4675:
4669:
4666:
4660:
4657:
4651:
4649:
4647:
4645:
4641:
4635:
4632:
4626:
4623:
4617:
4614:
4608:
4605:
4599:
4596:
4590:
4587:
4582:
4576:
4573:
4567:
4565:
4561:
4556:
4550:
4547:
4541:
4538:
4532:
4529:
4523:
4520:
4514:
4510:
4506:
4502:
4497:
4492:
4488:
4484:
4477:
4475:
4471:
4465:
4462:
4456:
4453:
4447:
4444:
4438:
4436:
4434:
4432:
4428:
4422:
4418:
4415:
4413:
4410:
4408:
4405:
4404:
4400:
4395:
4376:
4370:
4365:
4361:
4357:
4352:
4348:
4344:
4336:
4332:
4314:
4311:
4308:
4302:
4296:
4275:
4272:
4269:
4266:
4260:
4254:
4225:
4222:
4219:
4204:
4201:
4198:
4195:
4189:
4178:
4168:
4165:
4162:
4157:
4154:
4146:
4140:
4126:
4122:
4121:
4117:
4115:
4112:
4099:
4094:
4090:
4086:
4083:
4078:
4074:
4070:
4065:
4061:
4057:
4049:
4033:
4028:
4024:
4020:
4015:
4011:
4007:
3985:
3981:
3977:
3972:
3968:
3964:
3956:
3938:
3934:
3925:
3920:
3907:
3902:
3898:
3894:
3891:
3886:
3882:
3878:
3873:
3869:
3865:
3857:
3841:
3838:
3835:
3832:
3810:
3806:
3802:
3799:
3794:
3790:
3780:
3764:
3760:
3756:
3748:
3745:that contain
3744:
3740:
3722:
3718:
3714:
3706:
3704:
3700:
3696:
3678:
3674:
3670:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3618:
3613:
3610:
3606:
3602:
3594:
3590:
3582:
3565:
3560:
3510:
3504:
3477:
3435:
3383:
3343:
3292:
3274:
3271:
3263:
3260:
3257:belonging to
3256:
3240:
3237:
3231:
3220:
3217:
3198:
3195:
3192:
3186:
3183:
3163:
3160:
3157:
3149:
3148:
3144:
3142:
3128:
3123:
3119:
3115:
3112:
3107:
3103:
3080:
3076:
3056:
3053:
3050:
3047:
3038:
3034:
3030:
3025:
3011:
3008:
3002:
2994:
2990:
2986:
2982:
2978:
2974:
2969:
2956:
2953:
2950:
2947:
2925:
2921:
2917:
2912:
2908:
2904:
2896:
2893:that contain
2892:
2874:
2870:
2866:
2859:. As above,
2858:
2854:
2838:
2833:
2830:
2826:
2822:
2814:
2796:
2792:
2788:
2780:
2776:
2771:
2769:
2765:
2761:
2757:
2739:
2735:
2731:
2723:
2719:
2715:
2696:
2690:
2685:
2681:
2671:
2666:
2650:
2646:
2637:
2619:
2615:
2606:
2602:
2584:
2580:
2571:
2567:
2551:
2546:
2542:
2538:
2530:
2512:
2508:
2499:
2495:
2490:
2488:
2484:
2480:
2476:
2472:
2468:
2452:
2449:
2446:
2438:
2434:
2430:
2426:
2422:
2418:
2410:
2406:
2402:
2386:
2381:
2378:
2374:
2370:
2363:
2360:
2356:
2352:
2336:
2331:
2327:
2323:
2316:
2315:
2314:
2300:
2297:
2294:
2291:
2288:
2285:
2277:
2272:
2256:
2252:
2246:
2242:
2219:
2215:
2192:
2188:
2182:
2178:
2169:
2148:
2144:
2132:
2127:
2121:
2116:
2112:
2095:
2091:
2068:
2064:
2041:
2037:
2031:
2027:
2018:
2015:
2011:
2008:
2004:
1986:
1982:
1956:
1952:
1926:
1922:
1916:
1912:
1904:
1900:
1897:
1893:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1861:
1855:
1841:
1821:
1818:
1798:
1779:
1776:
1756:
1736:
1733:
1713:
1706:). Therefore
1693:
1673:
1653:
1634:
1631:
1611:
1591:
1588:
1581:contained in
1568:
1548:
1528:
1508:
1488:
1485:
1465:
1445:
1442:
1422:
1402:
1382:
1373:
1371:
1355:
1346:
1345:
1338:
1333:
1320:
1318:
1305:
1299:
1296:
1293:
1281:
1278:
1275:
1249:
1246:
1240:
1237:
1234:
1225:
1205:
1197:
1181:
1158:
1155:
1152:
1142:
1139:
1131:
1127:
1123:
1120:
1116:
1113:
1109:
1105:
1102:
1098:
1095:
1069:
1066:
1062:
1045:
1015:
1012:
1009:
1003:
997:
994:
991:
985:
982:
975:The subspace
974:
971:
967:
963:
946:
931:
927:
926:
922:
920:
905:
902:
882:
873:
859:
839:
819:
811:
798:
776:
767:
765:
749:
746:
726:
717:
715:
699:
679:
659:
651:
638:
628:
615:
593:
581:
579:
577:
572:
570:
554:
534:
526:
523:
507:
499:
486:
464:
455:
453:
426:
424:
420:
404:
384:
376:
373:
357:
349:
345:
332:
310:
301:
288:
285:
265:
245:
233:
231:
229:
225:
221:
217:
204:
202:
198:
194:
190:
186:
182:
177:
173:
171:
167:
163:
145:
130:
126:
122:
118:
114:
110:
106:
105:connectedness
98:
96:
94:
90:
86:
81:
79:
76:
72:
68:
64:
61:
57:
53:
45:
41:
37:
32:
19:
4889:
4885:
4858:
4854:
4808:
4782:
4746:
4725:
4716:
4707:
4686:
4677:
4668:
4659:
4634:
4625:
4616:
4607:
4598:
4589:
4575:
4549:
4540:
4531:
4522:
4486:
4482:
4464:
4455:
4446:
4393:
4113:
4047:
3954:
3923:
3921:
3855:
3781:
3746:
3742:
3707:
3702:
3698:
3694:
3660:
3656:
3652:
3648:
3644:
3640:
3636:
3632:
3592:
3588:
3586:
3258:
3254:
3036:
3032:
3028:
3026:
2992:
2988:
2984:
2980:
2976:
2972:
2970:
2894:
2890:
2856:
2852:
2812:
2778:
2774:
2772:
2767:
2763:
2759:
2755:
2721:
2713:
2669:
2667:
2635:
2604:
2600:
2569:
2528:
2497:
2493:
2491:
2482:
2474:
2470:
2466:
2436:
2432:
2428:
2424:
2420:
2416:
2414:
2408:
2404:
2400:
2358:
2354:
2350:
2275:
2273:
2167:
2130:
2128:
2125:
1941:of a family
1888:
1884:
1880:
1876:
1872:
1868:
1374:
1347:
1325:
1135:
929:
874:
790:
768:
764:broom spaces
718:
713:
630:
607:
585:
573:
568:
524:
478:
456:
427:
422:
374:
348:neighborhood
324:
302:
237:
224:Cantor space
205:
178:
174:
161:
128:
120:
102:
88:
84:
82:
66:
62:
49:
43:
39:
35:
4517:, section 2
3741:subsets of
3693:containing
2718:clopen sets
2665:is closed.
2603:containing
2129:Lemma: Let
1834:Therefore,
1811:is open in
1395:be open in
1126:Kirch space
520:contains a
370:contains a
234:Definitions
131:subsets of
109:compactness
56:mathematics
4919:Categories
4736:References
4127:The space
3782:Evidently
3643:such that
3358:(which is
3216:dictionary
2977:(x,sin(x))
2815:such that
2781:, the set
2773:Similarly
2531:such that
2500:, the set
1321:Properties
1101:comb space
1070:The space
606:is called
477:is called
323:is called
193:metrizable
99:Background
4807:(1995) ,
4513:255549682
4496:1311.5122
4303:×
4267:−
4261:×
4211:∖
4196:−
4190:×
4169:∈
4147:∪
4000:and thus
3978:⊆
3892:⊆
3879:⊆
3836:∈
3800:⊆
3607:≡
3561:ℓ
3505:ℓ
3384:). Since
3344:ℓ
3293:ℓ
3283:→
3238:×
3214:) in the
3161:×
3051:∈
3006:∖
2951:∈
2918:⊆
2827:≡
2732:∐
2543:≡
2450:∪
2375:≡
2328:≡
2295:∈
2243:⋃
2179:⋂
2028:∏
1913:∐
1300:τ
1288:→
1182:τ
1159:τ
1004:∪
372:connected
346:if every
181:manifolds
129:connected
78:connected
4783:Topology
4745:(1989).
4401:See also
4118:Examples
3825:for all
3150:The set
3145:Examples
2981:x > 0
2940:for all
1415:and let
769:A space
586:A space
457:A space
303:A space
228:discrete
214:(in the
52:topology
4908:2036067
4877:2037874
4833:1382863
3380:in the
3176:(where
2313:write:
1327:Theorem
121:compact
4906:
4875:
4831:
4821:
4789:
4774:
4753:
4511:
3739:clopen
3470:under
3428:under
2983:, and
2278:: for
2113:Every
1749:Since
1604:Since
970:convex
417:has a
95:sets.
80:sets.
4904:JSTOR
4873:JSTOR
4815:Dover
4509:S2CID
4491:arXiv
4423:Notes
3957:, so
2979:with
2485:into
2361:; and
1501:Then
1341:Proof
160:(for
4819:ISBN
4787:ISBN
4772:ISBN
4751:ISBN
4333:The
3651:and
3639:and
3587:Let
3264:Let
2991:are
2492:For
2473:and
2431:and
2427:and
2419:and
2407:and
2357:and
1901:The
1896:base
1458:Let
1375:Let
1124:The
1117:The
1099:The
1063:The
525:open
375:open
238:Let
107:and
75:open
58:, a
4894:doi
4863:doi
4501:doi
3922:If
3701:of
3546:to
3329:to
2855:of
2777:in
2758:of
2668:If
2568:of
2496:in
1883:in
1561:of
1198:on
1092:of
629:or
578:).
454:).
350:of
168:is
87:is
65:is
50:In
4921::
4902:,
4890:19
4888:,
4871:,
4859:32
4857:,
4829:MR
4827:,
4803:;
4770:;
4766:;
4695:^
4643:^
4563:^
4507:.
4499:.
4487:26
4485:.
4473:^
4430:^
3858::
3705:.
3595::
2469:,
1136:A
230:.
127:,
4911:.
4896::
4865::
4813:(
4796:.
4759:.
4583:.
4557:.
4515:.
4503::
4493::
4396:.
4394:x
4380:}
4377:x
4374:{
4371:=
4366:x
4362:C
4358:=
4353:x
4349:C
4345:Q
4318:]
4315:1
4312:,
4309:0
4306:(
4300:}
4297:0
4294:{
4279:)
4276:0
4273:,
4270:1
4264:[
4258:}
4255:0
4252:{
4232:}
4229:)
4226:0
4223:,
4220:0
4217:(
4214:{
4208:]
4205:1
4202:,
4199:1
4193:[
4187:)
4184:}
4179:+
4174:Z
4166:n
4163::
4158:n
4155:1
4150:{
4144:}
4141:0
4138:{
4135:(
4100:.
4095:x
4091:C
4087:Q
4084:=
4079:x
4075:C
4071:=
4066:x
4062:C
4058:P
4048:x
4034:.
4029:x
4025:C
4021:=
4016:x
4012:C
4008:Q
3986:x
3982:C
3973:x
3969:C
3965:Q
3955:x
3939:x
3935:C
3924:X
3908:.
3903:x
3899:C
3895:Q
3887:x
3883:C
3874:x
3870:C
3866:P
3856:x
3842:.
3839:X
3833:x
3811:x
3807:C
3803:Q
3795:x
3791:C
3765:x
3761:C
3757:Q
3747:x
3743:X
3723:x
3719:C
3715:Q
3703:x
3695:x
3679:x
3675:C
3671:Q
3661:X
3657:B
3653:y
3649:A
3645:x
3641:B
3637:A
3633:X
3619:y
3614:c
3611:q
3603:x
3593:X
3589:X
3566:,
3556:R
3533:R
3511:/
3500:R
3478:f
3457:R
3436:f
3415:R
3393:R
3367:R
3339:R
3316:R
3288:R
3279:R
3275::
3272:f
3261:.
3259:I
3255:a
3241:I
3235:}
3232:a
3229:{
3202:]
3199:1
3196:,
3193:0
3190:[
3187:=
3184:I
3164:I
3158:I
3129:.
3124:x
3120:C
3116:P
3113:=
3108:x
3104:C
3081:x
3077:C
3057:,
3054:X
3048:x
3037:X
3033:U
3029:U
3012:,
3009:U
3003:C
2993:U
2989:C
2985:U
2973:U
2957:.
2954:X
2948:x
2926:x
2922:C
2913:x
2909:C
2905:P
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2891:X
2875:x
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2867:P
2857:x
2839:x
2834:c
2831:p
2823:y
2813:y
2797:x
2793:C
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2779:X
2775:x
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2764:U
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2756:U
2740:x
2736:C
2722:X
2714:x
2700:}
2697:x
2694:{
2691:=
2686:x
2682:C
2670:X
2651:x
2647:C
2636:x
2620:x
2616:C
2605:x
2601:X
2585:x
2581:C
2570:x
2552:x
2547:c
2539:y
2529:y
2513:x
2509:C
2498:X
2494:x
2483:X
2475:z
2471:y
2467:x
2453:B
2447:A
2437:B
2433:z
2429:y
2425:A
2421:y
2417:x
2411:.
2409:y
2405:x
2401:X
2387:y
2382:c
2379:p
2371:x
2359:y
2355:x
2351:X
2337:y
2332:c
2324:x
2301:,
2298:X
2292:y
2289:,
2286:x
2276:X
2257:i
2253:Y
2247:i
2220:i
2216:Y
2193:i
2189:Y
2183:i
2168:X
2154:}
2149:i
2145:Y
2141:{
2131:X
2096:i
2092:X
2069:i
2065:X
2042:i
2038:X
2032:i
1987:i
1983:X
1962:}
1957:i
1953:X
1949:{
1927:i
1923:X
1917:i
1889:P
1885:X
1881:x
1877:P
1873:X
1869:P
1842:X
1822:.
1819:X
1799:C
1780:,
1777:C
1757:x
1737:.
1734:C
1714:x
1694:x
1674:C
1654:V
1635:,
1632:x
1612:V
1592:.
1589:U
1569:x
1549:V
1529:x
1509:U
1489:.
1486:C
1466:x
1446:.
1443:U
1423:C
1403:X
1383:U
1356:X
1306:.
1303:)
1297:,
1294:X
1291:(
1285:]
1282:1
1279:,
1276:0
1273:[
1253:)
1250:X
1247:;
1244:]
1241:1
1238:,
1235:0
1232:[
1229:(
1226:C
1206:X
1162:)
1156:,
1153:X
1150:(
1132:.
1079:Q
1046:1
1041:R
1019:]
1016:3
1013:,
1010:2
1007:[
1001:]
998:1
995:,
992:0
989:[
986:=
983:S
947:n
942:R
930:n
906:.
903:x
883:x
860:x
840:x
820:x
799:x
777:X
750:.
747:x
727:x
700:x
680:x
660:x
639:x
616:x
594:X
555:x
535:x
508:x
487:x
465:X
437:R
405:x
385:x
358:x
333:x
311:X
289:.
286:X
266:x
246:X
212:U
208:U
162:n
146:n
141:R
85:X
63:X
46:.
44:p
40:p
36:V
20:)
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