47:
6377:
5713:
4160:
3901:
5333:
6160:
6398:
2743:
6366:
6435:
6408:
6388:
5602:
As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected
5266:
4113:
3566:
Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small
935:
of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the
5681:
is, by definition, also required to be path connected, any simply connected space is also connected. If the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be
5088:
3908:
Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two
5548:
have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The
2136:
is connected, but is not simply connected. The three-dimensional
Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not
4527:
4463:
2722:
An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an
2545:
real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast,
1444:
2583:
2503:
1138:
1083:
4982:
3835:
if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about
4395:
1247:
3670:
3619:
1031:
4232:
3995:
3477:; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable.
1350:
5419:
4149:
3891:
3862:
3336:
3307:
2311:
2227:
2088:
3983:
2052:
982:
4652:
1195:
4756:
4730:
3933:
3278:
3248:
3160:
2370:
2348:
2198:
2162:
1735:
4923:
4863:
4574:
1277:
877:
807:
4696:
4331:
2694:
2256:
5513:
of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected).
4604:
3503:
3431:
2282:
594:
562:
5597:
2974:
2939:
2134:
1893:
1858:
5083:
5056:
5029:
4810:
4783:
3216:
1954:
1823:
1784:
1170:
5468:
1401:
1320:
850:
787:
681:
6438:
5668:
5648:
5571:
5439:
5387:
5367:
5326:
5306:
5286:
5002:
4883:
4830:
4351:
4298:
4275:
4255:
3822:
3802:
3774:
3754:
3713:
3690:
3639:
3588:
3556:
3533:
3475:
3455:
3368:
3183:
3130:
3110:
3090:
3070:
3050:
3026:
3002:
2904:
2849:
2829:
2806:
2786:
2714:
2662:
2640:
2613:
2543:
2523:
2450:
2426:
2007:
1713:
1693:
1673:
1653:
1633:
1613:
1593:
1570:
1550:
1530:
1502:
1464:
1374:
1297:
925:
901:
827:
764:
744:
724:
701:
658:
622:
530:
499:
477:
453:
430:
406:
383:
360:
325:
297:
258:
226:
194:
2884:
1927:
5261:{\displaystyle X=\left(Y\cup X_{1}\right)\cup X_{2}=\left(Z_{1}\cup Z_{2}\right)\cup X_{2}=\left(Z_{1}\cup X_{2}\right)\cup \left(Z_{2}\cap X_{1}\right)}
3188:
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended
6464:
1741:, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even
1715:. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers
5610:
However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see
3165:) if there is exactly one path-component. For non-empty spaces, this is equivalent to the statement that there is a path joining any two points in
5607:). Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.
2619:
5614:). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.
5506:
The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
6072:
6016:
5726:
3695:
If arc-connected subsets have a non-empty intersection, then their union may not be arc-connected. For example, the arc-components of
6426:
6421:
5968:
5878:
5850:
5523:
Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
4471:
4407:
46:
6416:
5740:
6318:
6008:
5746:
3379:
3338:
are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for
5693:
In general, any path connected space must be connected but there exist connected spaces that are not path connected. The
2548:
2468:
5545:
335:
of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the
4237:
Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets
6003:
5493:
of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected.
1406:
1088:
3562:. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces:
1036:
6326:
5698:
5471:
3989:
3219:
2456:
2380:
2376:
1475:
6459:
5766:
5510:
4928:
4533:
4333:
can be partitioned to two sub-collections, such that the unions of the sub-collections are disjoint and open in
6397:
6125:
5611:
3339:
2384:
2317:
4465:) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected.
4364:
2585:
is connected. More generally, the set of invertible bounded operators on a complex
Hilbert space is connected.
3644:
Arc-components of a product space may not be products of arc-components of the marginal spaces. For example,
6411:
5752:
3732:
3725:
2405:
is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
2391:
1377:
941:
264:
1745:, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
1200:
6346:
6341:
6267:
6144:
6132:
6105:
6065:
5678:
4108:{\displaystyle T=\{(0,0)\}\cup \left\{\left(x,\sin \left({\tfrac {1}{x}}\right)\right):x\in (0,1]\right\}}
3647:
3596:
3255:
987:
236:
170:
109:
6188:
6115:
4159:
3559:
3512:
3387:
2723:
6376:
5712:
5533:
Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected
4181:
5392:
4125:
3867:
3838:
3312:
3283:
2287:
2203:
2064:
6336:
6288:
6262:
6110:
5998:
5671:
3938:
3894:
3485:
3189:
3029:
2463:
2172:
2140:
A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
2091:
2013:
947:
6387:
6183:
5735:
5332:
4613:
3900:
2731:
2395:
1175:
509:
456:
113:
4735:
4709:
3916:
3261:
3231:
3143:
2353:
2331:
2200:, the remainder is disconnected. However, if even a countable infinity of points are removed from
2181:
2145:
1718:
1325:
6381:
6331:
6252:
6242:
6120:
6100:
5809:
5772:
5718:
5686:
5490:
4895:
4835:
4538:
4116:
2727:
1255:
904:
604:
Historically this modern formulation of the notion of connectedness (in terms of no partition of
339:(with its unique topology) as a connected space, but this article does not follow that practice.
156:
6351:
862:
792:
4657:
4303:
3511:
is closed. An example of a space which is path-connected but not arc-connected is given by the
6369:
6235:
6193:
6058:
5979:
5964:
5874:
5866:
5846:
5623:
3005:
2809:
2670:
2232:
1790:
1738:
597:
277:
240:
205:
152:
4583:
3488:
3392:
2459:
is an example of a set that is connected but is neither path connected nor locally connected.
2261:
567:
535:
6149:
6095:
5801:
5576:
5517:
4119:
2944:
2909:
2104:
1863:
1828:
6031:
5520:
of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
5061:
5034:
5007:
4788:
4761:
3538:
Intuition for path-connected spaces does not readily transfer to arc-connected spaces. Let
3194:
1900:
1796:
1757:
1143:
6208:
5444:
3781:
3508:
3481:
3383:
2764:
2098:
1787:
1742:
937:
853:
629:
166:
3988:
A classical example of a connected space that is not locally connected is the so-called
1383:
1302:
832:
769:
663:
6298:
6230:
5936:
5915:
5893:
5653:
5633:
5556:
5424:
5372:
5352:
5311:
5291:
5271:
4987:
4868:
4815:
4703:
4398:
4336:
4283:
4260:
4240:
3910:
3807:
3787:
3759:
3739:
3698:
3675:
3624:
3593:
Arc-connected product space may not be a product of arc-connected spaces. For example,
3573:
3541:
3518:
3460:
3440:
3353:
3251:
3168:
3115:
3095:
3075:
3055:
3035:
3011:
2987:
2889:
2834:
2814:
2791:
2771:
2699:
2647:
2625:
2598:
2528:
2508:
2435:
2411:
1959:
1698:
1678:
1658:
1638:
1618:
1598:
1578:
1555:
1535:
1515:
1487:
1449:
1359:
1282:
910:
886:
812:
749:
729:
709:
686:
643:
607:
515:
502:
484:
462:
438:
415:
391:
368:
345:
310:
282:
243:
211:
179:
2857:
6453:
6308:
6218:
6198:
5956:
2852:
2750:² is path-connected, because a path can be drawn between any two points in the space.
2321:
932:
928:
625:
6401:
4925:
is not connected. So it can be written as the union of two disjoint open sets, e.g.
4468:
If the sets can be ordered as a "linked chain", i.e. indexed by integer indices and
2742:
1956:
is disconnected; both of these intervals are open in the standard topological space
6293:
6213:
6159:
2390:
On the other hand, a finite set might be connected. For example, the spectrum of a
2325:
160:
5982:
385:
is connected, that is, it cannot be divided into two disjoint non-empty open sets.
5897:
2592:
and integral domains are connected. More generally, the following are equivalent
6391:
6303:
5758:
5550:
5486:
A locally path-connected space is path-connected if and only if it is connected.
3484:
that is path-connected is also arc-connected; more generally this is true for a
2165:
144:
6247:
6178:
6137:
5708:
5694:
5497:
5480:
In a locally path-connected space, every open connected set is path-connected.
4178:
of connected sets is not necessarily connected, as can be seen by considering
2665:
2589:
2402:
2058:
1737:, and identify them at every point except zero. The resulting space, with the
409:
17:
4401:. Hence the union of connected sets with non-empty intersection is connected.
4353:(see picture). This implies that in several cases, a union of connected sets
6272:
5987:
5792:
Wilder, R.L. (1978). "Evolution of the
Topological Concept of "Connected"".
5004:
is connected, it must be entirely contained in one of these components, say
3226:
2762:
is a stronger notion of connectedness, requiring the structure of a path. A
1573:
336:
163:
5827:
5470:
is (path-)connected. This result can be considered a generalization of the
5931:
5910:
5630:
If there exist no two disjoint non-empty open sets in a topological space
683:
the union of any collection of connected subsets such that each contained
624:
into two separated sets) first appeared (independently) with N.J. Lennes,
6257:
6225:
6174:
6081:
5911:"How to prove this result involving the quotient maps and connectedness?"
5527:
3904:
The topologist's sine curve is connected, but it is not locally connected
3804:
is locally connected if and only if every component of every open set of
2429:
140:
5749: – Topological space in which the closure of every open set is open
5813:
1793:
is connected; although it can, for example, be written as the union of
5689:. Every contractible space is path connected and thus also connected.
4758:
is disconnected (and thus can be written as a union of two open sets
1353:
931:, non-empty and their union is the whole space. Every component is a
857:
332:
77:
5805:
5503:
The connected components of a locally connected space are also open.
2379:
with at least two elements is disconnected, in fact such a space is
307:
if it is the union of two disjoint non-empty open sets. Otherwise,
5331:
4158:
2741:
5536:
Continuous image of arc-wise connected set is arc-wise connected.
4397:), then obviously they cannot be partitioned to collections with
2313:
remains simply connected after removal of countably many points.
944:), which are not open. Proof: Any two distinct rational numbers
6054:
3893:, each of which is locally path-connected. More generally, any
2394:
consists of two points and is connected. It is an example of a
5828:"General topology - Components of the set of rational numbers"
5685:
Yet stronger versions of connectivity include the notion of a
1474:
A space in which all components are one-point sets is called
5729: – Maximal subgraph whose vertices can reach each other
4706:
of connected sets is not necessarily connected. However, if
6050:
505:(sets for which each is disjoint from the other's closure).
4522:{\displaystyle \forall i:X_{i}\cap X_{i+1}\neq \emptyset }
4458:{\displaystyle \forall i,j:X_{i}\cap X_{j}\neq \emptyset }
632:
at the beginning of the 20th century. See for details.
5483:
Every locally path-connected space is locally connected.
5268:
The two sets in the last union are disjoint and open in
268:, which neither implies nor follows from connectedness.
5731:
Pages displaying short descriptions of redirect targets
5697:
furnishes such an example, as does the above-mentioned
4404:
If the intersection of each pair of sets is not empty (
984:
are in different components. Take an irrational number
4367:
4163:
Examples of unions and intersections of connected sets
4056:
883:
of the space. The components of any topological space
6024:
Mem. Fac. Sci. Eng. Shimane Univ., Series B: Math. Sc
5656:
5636:
5612:
topological graph theory#Graphs as topological spaces
5579:
5559:
5447:
5427:
5395:
5375:
5355:
5314:
5294:
5274:
5091:
5064:
5037:
5010:
4990:
4931:
4898:
4871:
4838:
4818:
4791:
4764:
4738:
4712:
4660:
4616:
4586:
4541:
4474:
4410:
4361:
If the common intersection of all sets is not empty (
4339:
4306:
4286:
4263:
4243:
4184:
4128:
3998:
3941:
3919:
3870:
3841:
3810:
3790:
3762:
3742:
3701:
3678:
3650:
3627:
3599:
3576:
3544:
3521:
3491:
3463:
3443:
3395:
3356:
3315:
3286:
3264:
3234:
3197:
3171:
3146:
3118:
3098:
3078:
3058:
3038:
3014:
2990:
2947:
2912:
2892:
2860:
2837:
2817:
2794:
2774:
2726:
removed, as well as the union of two disjoint closed
2702:
2673:
2650:
2628:
2601:
2551:
2531:
2511:
2471:
2438:
2414:
2356:
2334:
2290:
2264:
2235:
2206:
2184:
2148:
2107:
2067:
2016:
1962:
1930:
1903:
1866:
1860:
the second set is not open in the chosen topology of
1831:
1799:
1760:
1721:
1701:
1681:
1661:
1641:
1621:
1601:
1581:
1558:
1538:
1518:
1490:
1452:
1409:
1386:
1362:
1328:
1305:
1285:
1258:
1203:
1178:
1146:
1091:
1039:
990:
950:
913:
889:
865:
835:
815:
795:
772:
752:
732:
712:
689:
666:
646:
610:
570:
538:
518:
487:
465:
441:
418:
394:
371:
348:
313:
285:
246:
214:
182:
5743: – Connected open subset of a topological space
5336:
Two connected sets whose difference is not connected
6317:
6281:
6167:
6088:
2716:
is not a product of two rings in a nontrivial way).
2578:{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
2498:{\displaystyle \operatorname {GL} (n,\mathbb {R} )}
5662:
5642:
5591:
5565:
5462:
5433:
5413:
5381:
5361:
5320:
5300:
5280:
5260:
5077:
5050:
5023:
4996:
4976:
4917:
4877:
4857:
4824:
4804:
4777:
4750:
4724:
4690:
4646:
4598:
4568:
4521:
4457:
4389:
4345:
4325:
4292:
4269:
4249:
4226:
4143:
4107:
3977:
3927:
3885:
3856:
3816:
3796:
3768:
3748:
3707:
3684:
3664:
3633:
3613:
3582:
3550:
3527:
3497:
3469:
3449:
3425:
3362:
3330:
3301:
3272:
3242:
3210:
3177:
3154:
3124:
3104:
3084:
3064:
3044:
3020:
2996:
2968:
2933:
2898:
2878:
2843:
2823:
2800:
2780:
2708:
2688:
2656:
2634:
2607:
2577:
2537:
2517:
2497:
2444:
2420:
2364:
2342:
2305:
2276:
2250:
2221:
2192:
2156:
2128:
2082:
2046:
2001:
1948:
1921:
1887:
1852:
1817:
1778:
1729:
1707:
1687:
1667:
1647:
1627:
1607:
1587:
1564:
1544:
1524:
1496:
1458:
1438:
1395:
1368:
1344:
1314:
1291:
1271:
1241:
1189:
1164:
1132:
1077:
1025:
976:
919:
895:
879:) of a non-empty topological space are called the
871:
844:
821:
801:
781:
758:
738:
718:
695:
675:
652:
616:
588:
556:
524:
493:
471:
447:
424:
400:
377:
354:
319:
291:
252:
220:
188:
5898:The K-book: An introduction to algebraic K-theory
3570:Arc-components may not be disjoint. For example,
173:that are used to distinguish topological spaces.
4171:of connected sets is not necessarily connected.
3784:of connected sets. It can be shown that a space
3715:intersect, but their union is not arc-connected.
3185:. Again, many authors exclude the empty space.
2730:, where all examples of this paragraph bear the
501:cannot be written as the union of two non-empty
5932:"How to prove this result about connectedness?"
3776:contains a connected open neighbourhood. It is
3254:they are path-connected; these subsets are the
1439:{\displaystyle \Gamma _{x}\subset \Gamma '_{x}}
1133:{\displaystyle B=\{q\in \mathbb {Q} :q>r\}.}
5622:There are stronger forms of connectedness for
5604:
3535:can be connected by a path but not by an arc.
1078:{\displaystyle A=\{q\in \mathbb {Q} :q<r\}}
6066:
4698:are disjoint and open in the quotient space).
4654:is a separation of the quotient space (since
3827:Similarly, a topological space is said to be
8:
5871:Introduction to Topology and Modern Analysis
4832:with each such component is connected (i.e.
4563:
4550:
4320:
4307:
4023:
4005:
2734:induced by two-dimensional Euclidean space.
2041:
2035:
1124:
1098:
1072:
1046:
703:will once again be a connected subset. The
583:
571:
551:
539:
204:if it is a connected space when viewed as a
4977:{\displaystyle Y\cup X_{1}=Z_{1}\cup Z_{2}}
4529:, then again their union must be connected.
1466:is compact Hausdorff or locally connected.
6434:
6407:
6073:
6059:
6051:
4532:If the sets are pairwise-disjoint and the
1249:. Thus each component is a one-point set.
789:it is the unique largest (with respect to
5873:. McGraw Hill Book Company. p. 144.
5655:
5635:
5578:
5558:
5446:
5426:
5394:
5374:
5354:
5313:
5293:
5273:
5247:
5234:
5211:
5198:
5180:
5162:
5149:
5131:
5113:
5090:
5069:
5063:
5042:
5036:
5015:
5009:
4989:
4968:
4955:
4942:
4930:
4909:
4897:
4870:
4849:
4837:
4817:
4796:
4790:
4769:
4763:
4737:
4711:
4659:
4615:
4585:
4557:
4545:
4540:
4501:
4488:
4473:
4443:
4430:
4409:
4390:{\textstyle \bigcap X_{i}\neq \emptyset }
4375:
4366:
4338:
4314:
4305:
4285:
4262:
4242:
4183:
4135:
4131:
4130:
4127:
4055:
3997:
3940:
3921:
3920:
3918:
3877:
3873:
3872:
3869:
3848:
3844:
3843:
3840:
3809:
3789:
3761:
3741:
3700:
3677:
3658:
3657:
3649:
3626:
3607:
3606:
3598:
3575:
3543:
3520:
3507:, which is a space where each image of a
3490:
3462:
3442:
3394:
3355:
3322:
3318:
3317:
3314:
3293:
3289:
3288:
3285:
3266:
3265:
3263:
3236:
3235:
3233:
3202:
3196:
3170:
3147:
3145:
3117:
3097:
3077:
3057:
3037:
3013:
2989:
2946:
2911:
2891:
2859:
2836:
2816:
2793:
2773:
2701:
2672:
2649:
2627:
2600:
2568:
2567:
2550:
2530:
2510:
2488:
2487:
2470:
2437:
2413:
2358:
2357:
2355:
2336:
2335:
2333:
2297:
2293:
2292:
2289:
2263:
2234:
2213:
2209:
2208:
2205:
2186:
2185:
2183:
2150:
2149:
2147:
2106:
2074:
2070:
2069:
2066:
2015:
1961:
1929:
1902:
1865:
1830:
1798:
1759:
1723:
1722:
1720:
1700:
1680:
1660:
1640:
1620:
1600:
1580:
1557:
1537:
1517:
1489:
1451:
1427:
1414:
1408:
1385:
1361:
1333:
1327:
1304:
1284:
1263:
1257:
1227:
1208:
1202:
1180:
1179:
1177:
1145:
1108:
1107:
1090:
1056:
1055:
1038:
1014:
995:
989:
968:
955:
949:
912:
888:
864:
834:
814:
794:
771:
751:
746:is the union of all connected subsets of
731:
711:
688:
665:
645:
609:
569:
537:
517:
486:
464:
440:
417:
393:
370:
362:the following conditions are equivalent:
347:
312:
284:
245:
231:Some related but stronger conditions are
213:
181:
5477:Every path-connected space is connected.
3899:
596:is the two-point space endowed with the
169:. Connectedness is one of the principal
35:Connected and disconnected subspaces of
5784:
4742:
2178:If even a single point is removed from
232:
5755: – Property of topological spaces
4357:necessarily connected. In particular:
2168:with the usual topology, is connected.
5909:Brandsma, Henno (February 13, 2013).
4300:is disconnected, then the collection
3457:is a maximal arc-connected subset of
1242:{\displaystyle q_{1}\in A,q_{2}\in B}
7:
5496:The connected components are always
3665:{\displaystyle X\times \mathbb {R} }
3614:{\displaystyle X\times \mathbb {R} }
2620:finitely generated projective module
1484:. Related to this property, a space
1026:{\displaystyle q_{1}<r<q_{2},}
5441:is (path-)connected then the image
3590:has two overlapping arc-components.
2595:The spectrum of a commutative ring
27:Topological space that is connected
5727:Connected component (graph theory)
4516:
4475:
4452:
4411:
4384:
3730:A topological space is said to be
3492:
2328:, over a connected field (such as
1512:if, for any two distinct elements
1424:
1411:
1330:
1260:
155:that cannot be represented as the
25:
4580:must be connected. Otherwise, if
4227:{\displaystyle X=(0,1)\cup (1,2)}
6465:Properties of topological spaces
6433:
6406:
6396:
6386:
6375:
6365:
6364:
6158:
5711:
5414:{\displaystyle f:X\rightarrow Y}
4144:{\displaystyle \mathbb {R} ^{2}}
3886:{\displaystyle \mathbb {C} ^{n}}
3857:{\displaystyle \mathbb {R} ^{n}}
3672:has a single arc-component, but
3331:{\displaystyle \mathbb {C} ^{n}}
3302:{\displaystyle \mathbb {R} ^{n}}
3148:
2306:{\displaystyle \mathbb {R} ^{n}}
2222:{\displaystyle \mathbb {R} ^{n}}
2083:{\displaystyle \mathbb {R} ^{n}}
408:which are both open and closed (
176:A subset of a topological space
45:
6015:Muscat, J; Buhagiar, D (2006).
5618:Stronger forms of connectedness
3978:{\displaystyle (0,1)\cup (2,3)}
2383:. The simplest example is the
2258:the remainder is connected. If
2047:{\displaystyle (0,1)\cup \{3\}}
5741:Domain (mathematical analysis)
5457:
5451:
5405:
5389:be topological spaces and let
5308:, contradicting the fact that
5288:, so there is a separation of
4685:
4679:
4670:
4664:
4641:
4635:
4626:
4620:
4280:This means that, if the union
4221:
4209:
4203:
4191:
4097:
4085:
4020:
4008:
3972:
3960:
3954:
3942:
3417:
3414:
3402:
2957:
2951:
2922:
2916:
2873:
2861:
2572:
2558:
2492:
2478:
2120:
2108:
2029:
2017:
1993:
1981:
1975:
1963:
1943:
1931:
1916:
1904:
1879:
1867:
1844:
1832:
1812:
1800:
1773:
1761:
1279:be the connected component of
1159:
1147:
977:{\displaystyle q_{1}<q_{2}}
856:connected subsets (ordered by
705:connected component of a point
52:From top to bottom: red space
1:
5794:American Mathematical Monthly
5769: – Type of uniform space
5747:Extremally disconnected space
5421:be a continuous function. If
5347:Main theorem of connectedness
4647:{\displaystyle q(U)\cup q(V)}
3380:topologically distinguishable
2090:is connected; it is actually
1190:{\displaystyle \mathbb {Q} ,}
5670:must be connected, and thus
4751:{\displaystyle X\setminus Y}
4725:{\displaystyle X\supseteq Y}
3928:{\displaystyle \mathbb {R} }
3273:{\displaystyle \mathbb {R} }
3243:{\displaystyle \mathbb {R} }
3155:{\displaystyle \mathbf {0} }
2365:{\displaystyle \mathbb {C} }
2343:{\displaystyle \mathbb {R} }
2193:{\displaystyle \mathbb {R} }
2157:{\displaystyle \mathbb {R} }
1730:{\displaystyle \mathbb {Q} }
1446:where the equality holds if
1345:{\displaystyle \Gamma _{x}'}
262:. Another related notion is
6004:Encyclopedia of Mathematics
5930:Marek (February 13, 2013).
4918:{\displaystyle Y\cup X_{1}}
4858:{\displaystyle Y\cup X_{i}}
4569:{\displaystyle X/\{X_{i}\}}
3897:is locally path-connected.
3382:points can be joined by an
2588:The spectra of commutative
2432:to a connected space, then
1352:be the intersection of all
1272:{\displaystyle \Gamma _{x}}
6481:
6327:Banach fixed-point theorem
5605:Muscat & Buhagiar 2006
5599:odd) is one such example.
5530:is locally path-connected.
5472:intermediate value theorem
4892:By contradiction, suppose
3756:if every neighbourhood of
3723:
2377:discrete topological space
872:{\displaystyle \subseteq }
802:{\displaystyle \subseteq }
6360:
6156:
5767:Uniformly connected space
5500:(but in general not open)
4691:{\displaystyle q(U),q(V)}
4326:{\displaystyle \{X_{i}\}}
3340:finite topological spaces
2831:is a continuous function
5997:V. I. Malykhin (2001) ,
5961:Topology, Second Edition
5841:Stephen Willard (1970).
5603:sets to connected sets (
3280:. Also, open subsets of
3072:if there is a path from
2689:{\displaystyle \neq 0,1}
2385:discrete two-point space
2318:topological vector space
2251:{\displaystyle n\geq 2,}
940:are the one-point sets (
342:For a topological space
143:and related branches of
5753:Locally connected space
5699:topologist's sine curve
4599:{\displaystyle U\cup V}
3990:topologist's sine curve
3726:Locally connected space
3692:has two arc-components.
3498:{\displaystyle \Delta }
3426:{\displaystyle f:\to X}
3220:topologist's sine curve
2505:(that is, the group of
2457:topologist's sine curve
2392:discrete valuation ring
2372:), is simply connected.
2277:{\displaystyle n\geq 3}
1572:, there exist disjoint
1299:in a topological space
660:in a topological space
589:{\displaystyle \{0,1\}}
557:{\displaystyle \{0,1\}}
6382:Mathematics portal
6282:Metrics and properties
6268:Second-countable space
6030:: 1–13. Archived from
5845:. Dover. p. 191.
5679:simply connected space
5664:
5644:
5593:
5592:{\displaystyle n>3}
5567:
5464:
5435:
5415:
5383:
5363:
5337:
5322:
5302:
5282:
5262:
5079:
5052:
5025:
4998:
4978:
4919:
4879:
4859:
4826:
4806:
4779:
4752:
4726:
4692:
4648:
4600:
4570:
4523:
4459:
4391:
4347:
4327:
4294:
4271:
4251:
4228:
4164:
4145:
4109:
3979:
3929:
3905:
3887:
3858:
3831:locally path-connected
3818:
3798:
3770:
3750:
3709:
3686:
3666:
3635:
3621:is arc-connected, but
3615:
3584:
3552:
3529:
3499:
3471:
3451:
3427:
3364:
3332:
3303:
3274:
3244:
3212:
3179:
3156:
3126:
3106:
3086:
3066:
3046:
3022:
2998:
2970:
2969:{\displaystyle f(1)=y}
2935:
2934:{\displaystyle f(0)=x}
2900:
2880:
2845:
2825:
2802:
2782:
2751:
2710:
2690:
2658:
2636:
2609:
2579:
2539:
2519:
2499:
2446:
2422:
2366:
2344:
2307:
2278:
2252:
2223:
2194:
2158:
2130:
2129:{\displaystyle (0,0),}
2101:excluding the origin,
2084:
2048:
2003:
1950:
1923:
1889:
1888:{\displaystyle [0,2).}
1854:
1853:{\displaystyle [1,2),}
1819:
1780:
1731:
1709:
1689:
1669:
1649:
1629:
1609:
1589:
1566:
1546:
1526:
1498:
1460:
1440:
1397:
1370:
1346:
1316:
1293:
1273:
1243:
1191:
1166:
1134:
1079:
1027:
978:
921:
897:
873:
846:
823:
809:) connected subset of
803:
783:
760:
740:
720:
697:
677:
654:
618:
590:
558:
526:
495:
473:
449:
426:
402:
379:
356:
321:
293:
254:
222:
190:
171:topological properties
72:, whereas green space
5672:hyperconnected spaces
5665:
5645:
5594:
5568:
5465:
5436:
5416:
5384:
5364:
5335:
5323:
5303:
5283:
5263:
5080:
5078:{\displaystyle X_{1}}
5053:
5051:{\displaystyle Z_{2}}
5026:
5024:{\displaystyle Z_{1}}
4999:
4979:
4920:
4880:
4865:is connected for all
4860:
4827:
4812:), then the union of
4807:
4805:{\displaystyle X_{2}}
4780:
4778:{\displaystyle X_{1}}
4753:
4732:and their difference
4727:
4693:
4649:
4601:
4571:
4524:
4460:
4392:
4348:
4328:
4295:
4272:
4252:
4229:
4162:
4146:
4110:
3980:
3930:
3903:
3888:
3859:
3819:
3799:
3771:
3751:
3710:
3687:
3667:
3636:
3616:
3585:
3560:line with two origins
3553:
3530:
3513:line with two origins
3500:
3472:
3452:
3428:
3365:
3333:
3304:
3275:
3245:
3213:
3211:{\displaystyle L^{*}}
3180:
3157:
3127:
3107:
3087:
3067:
3047:
3023:
2999:
2971:
2936:
2901:
2881:
2846:
2826:
2803:
2783:
2745:
2711:
2691:
2659:
2637:
2610:
2580:
2540:
2520:
2500:
2447:
2423:
2367:
2345:
2308:
2279:
2253:
2224:
2195:
2159:
2131:
2085:
2049:
2004:
1951:
1949:{\displaystyle (1,2]}
1924:
1922:{\displaystyle [0,1)}
1890:
1855:
1820:
1818:{\displaystyle [0,1)}
1781:
1779:{\displaystyle [0,2)}
1732:
1710:
1690:
1670:
1650:
1630:
1610:
1590:
1567:
1547:
1527:
1499:
1461:
1441:
1398:
1371:
1347:
1317:
1294:
1274:
1244:
1192:
1167:
1165:{\displaystyle (A,B)}
1135:
1080:
1028:
979:
922:
898:
874:
847:
824:
804:
784:
761:
741:
721:
698:
678:
655:
619:
591:
559:
527:
496:
474:
450:
427:
403:
380:
357:
322:
294:
255:
223:
191:
6337:Invariance of domain
6289:Euler characteristic
6263:Bundle (mathematics)
5654:
5634:
5577:
5557:
5463:{\displaystyle f(X)}
5445:
5425:
5393:
5373:
5353:
5312:
5292:
5272:
5089:
5085:. Now we know that:
5062:
5035:
5008:
4988:
4929:
4896:
4869:
4836:
4816:
4789:
4762:
4736:
4710:
4658:
4614:
4584:
4539:
4472:
4408:
4365:
4337:
4304:
4284:
4261:
4241:
4182:
4126:
3996:
3939:
3917:
3895:topological manifold
3868:
3839:
3808:
3788:
3760:
3740:
3699:
3676:
3648:
3625:
3597:
3574:
3542:
3519:
3515:; its two copies of
3489:
3461:
3441:
3393:
3354:
3313:
3284:
3262:
3232:
3195:
3169:
3144:
3116:
3096:
3076:
3056:
3036:
3030:equivalence relation
3012:
2988:
2945:
2910:
2890:
2858:
2835:
2815:
2792:
2772:
2758:path-connected space
2700:
2671:
2648:
2626:
2599:
2549:
2529:
2509:
2469:
2464:general linear group
2452:is itself connected.
2436:
2412:
2381:totally disconnected
2354:
2332:
2288:
2262:
2233:
2204:
2182:
2146:
2105:
2065:
2014:
1960:
1928:
1901:
1864:
1829:
1797:
1758:
1754:The closed interval
1719:
1699:
1679:
1659:
1639:
1619:
1599:
1579:
1556:
1536:
1516:
1488:
1479:totally disconnected
1450:
1407:
1384:
1360:
1326:
1303:
1283:
1256:
1201:
1176:
1144:
1089:
1037:
988:
948:
911:
887:
881:connected components
863:
833:
813:
793:
770:
750:
730:
710:
687:
664:
644:
636:Connected components
608:
568:
564:are constant, where
536:
516:
485:
463:
439:
435:The only subsets of
416:
392:
388:The only subsets of
369:
346:
311:
283:
244:
212:
180:
6347:Tychonoff's theorem
6342:Poincaré conjecture
6096:General (point-set)
6017:"Connective Spaces"
5736:Connectedness locus
5674:are also connected.
4606:is a separation of
4576:is connected, then
3720:Local connectedness
2430:homotopy equivalent
1470:Disconnected spaces
1435:
1341:
1172:is a separation of
6332:De Rham cohomology
6253:Polyhedral complex
6243:Simplicial complex
5980:Weisstein, Eric W.
5773:Pixel connectivity
5719:Mathematics portal
5695:deleted comb space
5687:contractible space
5660:
5640:
5624:topological spaces
5589:
5563:
5460:
5431:
5411:
5379:
5359:
5338:
5318:
5298:
5278:
5258:
5075:
5048:
5021:
4994:
4974:
4915:
4890:
4875:
4855:
4822:
4802:
4775:
4748:
4722:
4688:
4644:
4596:
4566:
4519:
4455:
4387:
4343:
4323:
4290:
4267:
4247:
4224:
4165:
4141:
4117:Euclidean topology
4105:
4065:
3975:
3925:
3906:
3883:
3854:
3814:
3794:
3766:
3746:
3705:
3682:
3662:
3631:
3611:
3580:
3548:
3525:
3495:
3467:
3447:
3423:
3360:
3328:
3299:
3270:
3240:
3208:
3175:
3152:
3138:pathwise connected
3122:
3102:
3082:
3062:
3042:
3018:
2994:
2966:
2931:
2896:
2876:
2841:
2821:
2798:
2778:
2752:
2738:Path connectedness
2706:
2686:
2654:
2642:has constant rank.
2632:
2605:
2575:
2535:
2515:
2495:
2442:
2418:
2362:
2340:
2303:
2274:
2248:
2219:
2190:
2154:
2126:
2080:
2044:
1999:
1946:
1919:
1885:
1850:
1815:
1776:
1727:
1705:
1685:
1665:
1645:
1625:
1605:
1585:
1562:
1542:
1522:
1494:
1456:
1436:
1423:
1396:{\displaystyle x.}
1393:
1366:
1342:
1329:
1315:{\displaystyle X,}
1312:
1289:
1269:
1239:
1187:
1162:
1130:
1075:
1023:
974:
917:
893:
869:
845:{\displaystyle x.}
842:
819:
799:
782:{\displaystyle x;}
779:
756:
736:
716:
693:
676:{\displaystyle X,}
673:
650:
614:
586:
554:
522:
491:
479:and the empty set.
469:
445:
432:and the empty set.
422:
398:
375:
352:
317:
289:
250:
218:
186:
6447:
6446:
6236:fundamental group
5999:"Connected space"
5963:. Prentice Hall.
5957:Munkres, James R.
5867:George F. Simmons
5663:{\displaystyle X}
5643:{\displaystyle X}
5626:, for instance:
5566:{\displaystyle n}
5434:{\displaystyle X}
5382:{\displaystyle Y}
5362:{\displaystyle X}
5321:{\displaystyle X}
5301:{\displaystyle X}
5281:{\displaystyle X}
4997:{\displaystyle Y}
4888:
4878:{\displaystyle i}
4825:{\displaystyle Y}
4346:{\displaystyle X}
4293:{\displaystyle X}
4270:{\displaystyle V}
4250:{\displaystyle U}
4064:
3817:{\displaystyle X}
3797:{\displaystyle X}
3778:locally connected
3769:{\displaystyle x}
3749:{\displaystyle x}
3733:locally connected
3708:{\displaystyle X}
3685:{\displaystyle X}
3634:{\displaystyle X}
3583:{\displaystyle X}
3551:{\displaystyle X}
3528:{\displaystyle 0}
3470:{\displaystyle X}
3450:{\displaystyle X}
3376:arcwise connected
3363:{\displaystyle X}
3346:Arc connectedness
3178:{\displaystyle X}
3125:{\displaystyle X}
3105:{\displaystyle y}
3085:{\displaystyle x}
3065:{\displaystyle y}
3045:{\displaystyle x}
3021:{\displaystyle X}
3006:equivalence class
2997:{\displaystyle X}
2899:{\displaystyle X}
2844:{\displaystyle f}
2824:{\displaystyle X}
2810:topological space
2801:{\displaystyle y}
2781:{\displaystyle x}
2746:This subspace of
2732:subspace topology
2709:{\displaystyle R}
2657:{\displaystyle R}
2635:{\displaystyle R}
2608:{\displaystyle R}
2538:{\displaystyle n}
2518:{\displaystyle n}
2445:{\displaystyle X}
2421:{\displaystyle X}
2002:{\displaystyle .}
1791:subspace topology
1739:quotient topology
1708:{\displaystyle V}
1688:{\displaystyle U}
1668:{\displaystyle X}
1648:{\displaystyle y}
1628:{\displaystyle V}
1608:{\displaystyle x}
1588:{\displaystyle U}
1565:{\displaystyle X}
1545:{\displaystyle y}
1525:{\displaystyle x}
1508:totally separated
1497:{\displaystyle X}
1459:{\displaystyle X}
1369:{\displaystyle x}
1292:{\displaystyle x}
920:{\displaystyle X}
896:{\displaystyle X}
822:{\displaystyle X}
759:{\displaystyle X}
739:{\displaystyle X}
719:{\displaystyle x}
696:{\displaystyle x}
653:{\displaystyle x}
640:Given some point
617:{\displaystyle X}
598:discrete topology
525:{\displaystyle X}
494:{\displaystyle X}
472:{\displaystyle X}
448:{\displaystyle X}
425:{\displaystyle X}
401:{\displaystyle X}
378:{\displaystyle X}
355:{\displaystyle X}
320:{\displaystyle X}
292:{\displaystyle X}
278:topological space
272:Formal definition
265:locally connected
253:{\displaystyle n}
221:{\displaystyle X}
189:{\displaystyle X}
153:topological space
64:and orange space
16:(Redirected from
6472:
6460:General topology
6437:
6436:
6410:
6409:
6400:
6390:
6380:
6379:
6368:
6367:
6162:
6075:
6068:
6061:
6052:
6045:
6043:
6042:
6036:
6021:
6011:
5993:
5992:
5974:
5942:
5941:
5927:
5921:
5920:
5906:
5900:
5891:
5885:
5884:
5863:
5857:
5856:
5843:General Topology
5838:
5832:
5831:
5824:
5818:
5817:
5789:
5732:
5721:
5716:
5715:
5669:
5667:
5666:
5661:
5649:
5647:
5646:
5641:
5598:
5596:
5595:
5590:
5572:
5570:
5569:
5564:
5469:
5467:
5466:
5461:
5440:
5438:
5437:
5432:
5420:
5418:
5417:
5412:
5388:
5386:
5385:
5380:
5368:
5366:
5365:
5360:
5327:
5325:
5324:
5319:
5307:
5305:
5304:
5299:
5287:
5285:
5284:
5279:
5267:
5265:
5264:
5259:
5257:
5253:
5252:
5251:
5239:
5238:
5221:
5217:
5216:
5215:
5203:
5202:
5185:
5184:
5172:
5168:
5167:
5166:
5154:
5153:
5136:
5135:
5123:
5119:
5118:
5117:
5084:
5082:
5081:
5076:
5074:
5073:
5058:is contained in
5057:
5055:
5054:
5049:
5047:
5046:
5030:
5028:
5027:
5022:
5020:
5019:
5003:
5001:
5000:
4995:
4983:
4981:
4980:
4975:
4973:
4972:
4960:
4959:
4947:
4946:
4924:
4922:
4921:
4916:
4914:
4913:
4884:
4882:
4881:
4876:
4864:
4862:
4861:
4856:
4854:
4853:
4831:
4829:
4828:
4823:
4811:
4809:
4808:
4803:
4801:
4800:
4784:
4782:
4781:
4776:
4774:
4773:
4757:
4755:
4754:
4749:
4731:
4729:
4728:
4723:
4697:
4695:
4694:
4689:
4653:
4651:
4650:
4645:
4609:
4605:
4603:
4602:
4597:
4579:
4575:
4573:
4572:
4567:
4562:
4561:
4549:
4528:
4526:
4525:
4520:
4512:
4511:
4493:
4492:
4464:
4462:
4461:
4456:
4448:
4447:
4435:
4434:
4396:
4394:
4393:
4388:
4380:
4379:
4352:
4350:
4349:
4344:
4332:
4330:
4329:
4324:
4319:
4318:
4299:
4297:
4296:
4291:
4276:
4274:
4273:
4268:
4256:
4254:
4253:
4248:
4233:
4231:
4230:
4225:
4150:
4148:
4147:
4142:
4140:
4139:
4134:
4122:by inclusion in
4114:
4112:
4111:
4106:
4104:
4100:
4075:
4071:
4070:
4066:
4057:
3984:
3982:
3981:
3976:
3934:
3932:
3931:
3926:
3924:
3892:
3890:
3889:
3884:
3882:
3881:
3876:
3863:
3861:
3860:
3855:
3853:
3852:
3847:
3833:
3832:
3823:
3821:
3820:
3815:
3803:
3801:
3800:
3795:
3775:
3773:
3772:
3767:
3755:
3753:
3752:
3747:
3714:
3712:
3711:
3706:
3691:
3689:
3688:
3683:
3671:
3669:
3668:
3663:
3661:
3640:
3638:
3637:
3632:
3620:
3618:
3617:
3612:
3610:
3589:
3587:
3586:
3581:
3557:
3555:
3554:
3549:
3534:
3532:
3531:
3526:
3505:-Hausdorff space
3504:
3502:
3501:
3496:
3476:
3474:
3473:
3468:
3456:
3454:
3453:
3448:
3432:
3430:
3429:
3424:
3369:
3367:
3366:
3361:
3337:
3335:
3334:
3329:
3327:
3326:
3321:
3308:
3306:
3305:
3300:
3298:
3297:
3292:
3279:
3277:
3276:
3271:
3269:
3249:
3247:
3246:
3241:
3239:
3217:
3215:
3214:
3209:
3207:
3206:
3184:
3182:
3181:
3176:
3161:
3159:
3158:
3153:
3151:
3131:
3129:
3128:
3123:
3111:
3109:
3108:
3103:
3091:
3089:
3088:
3083:
3071:
3069:
3068:
3063:
3051:
3049:
3048:
3043:
3027:
3025:
3024:
3019:
3003:
3001:
3000:
2995:
2982:
2981:
2975:
2973:
2972:
2967:
2940:
2938:
2937:
2932:
2905:
2903:
2902:
2897:
2885:
2883:
2882:
2879:{\displaystyle }
2877:
2850:
2848:
2847:
2842:
2830:
2828:
2827:
2822:
2807:
2805:
2804:
2799:
2787:
2785:
2784:
2779:
2760:
2759:
2715:
2713:
2712:
2707:
2695:
2693:
2692:
2687:
2663:
2661:
2660:
2655:
2641:
2639:
2638:
2633:
2614:
2612:
2611:
2606:
2584:
2582:
2581:
2576:
2571:
2544:
2542:
2541:
2536:
2524:
2522:
2521:
2516:
2504:
2502:
2501:
2496:
2491:
2451:
2449:
2448:
2443:
2427:
2425:
2424:
2419:
2396:Sierpiński space
2371:
2369:
2368:
2363:
2361:
2349:
2347:
2346:
2341:
2339:
2312:
2310:
2309:
2304:
2302:
2301:
2296:
2283:
2281:
2280:
2275:
2257:
2255:
2254:
2249:
2228:
2226:
2225:
2220:
2218:
2217:
2212:
2199:
2197:
2196:
2191:
2189:
2175:is disconnected.
2163:
2161:
2160:
2155:
2153:
2135:
2133:
2132:
2127:
2092:simply connected
2089:
2087:
2086:
2081:
2079:
2078:
2073:
2054:is disconnected.
2053:
2051:
2050:
2045:
2008:
2006:
2005:
2000:
1955:
1953:
1952:
1947:
1926:
1925:
1920:
1894:
1892:
1891:
1886:
1859:
1857:
1856:
1851:
1824:
1822:
1821:
1816:
1785:
1783:
1782:
1777:
1736:
1734:
1733:
1728:
1726:
1714:
1712:
1711:
1706:
1694:
1692:
1691:
1686:
1675:is the union of
1674:
1672:
1671:
1666:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1594:
1592:
1591:
1586:
1571:
1569:
1568:
1563:
1551:
1549:
1548:
1543:
1531:
1529:
1528:
1523:
1510:
1509:
1503:
1501:
1500:
1495:
1481:
1480:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1431:
1419:
1418:
1402:
1400:
1399:
1394:
1375:
1373:
1372:
1367:
1356:sets containing
1351:
1349:
1348:
1343:
1337:
1321:
1319:
1318:
1313:
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1268:
1267:
1248:
1246:
1245:
1240:
1232:
1231:
1213:
1212:
1196:
1194:
1193:
1188:
1183:
1171:
1169:
1168:
1163:
1139:
1137:
1136:
1131:
1111:
1084:
1082:
1081:
1076:
1059:
1032:
1030:
1029:
1024:
1019:
1018:
1000:
999:
983:
981:
980:
975:
973:
972:
960:
959:
938:rational numbers
926:
924:
923:
918:
902:
900:
899:
894:
878:
876:
875:
870:
851:
849:
848:
843:
828:
826:
825:
820:
808:
806:
805:
800:
788:
786:
785:
780:
765:
763:
762:
757:
745:
743:
742:
737:
725:
723:
722:
717:
702:
700:
699:
694:
682:
680:
679:
674:
659:
657:
656:
651:
623:
621:
620:
615:
595:
593:
592:
587:
563:
561:
560:
555:
531:
529:
528:
523:
500:
498:
497:
492:
478:
476:
475:
470:
454:
452:
451:
446:
431:
429:
428:
423:
407:
405:
404:
399:
384:
382:
381:
376:
361:
359:
358:
353:
326:
324:
323:
318:
305:
304:
298:
296:
295:
290:
259:
257:
256:
251:
237:simply connected
227:
225:
224:
219:
202:
201:
195:
193:
192:
187:
128:has genus 1 and
110:simply connected
70:connected spaces
49:
21:
6480:
6479:
6475:
6474:
6473:
6471:
6470:
6469:
6450:
6449:
6448:
6443:
6374:
6356:
6352:Urysohn's lemma
6313:
6277:
6163:
6154:
6126:low-dimensional
6084:
6079:
6049:
6040:
6038:
6034:
6019:
6014:
5996:
5983:"Connected Set"
5978:
5977:
5971:
5955:
5951:
5949:Further reading
5946:
5945:
5929:
5928:
5924:
5908:
5907:
5903:
5892:
5888:
5881:
5865:
5864:
5860:
5853:
5840:
5839:
5835:
5826:
5825:
5821:
5806:10.2307/2321676
5791:
5790:
5786:
5781:
5730:
5717:
5710:
5707:
5652:
5651:
5632:
5631:
5620:
5575:
5574:
5555:
5554:
5553:graph (and any
5543:
5443:
5442:
5423:
5422:
5391:
5390:
5371:
5370:
5351:
5350:
5343:
5330:
5310:
5309:
5290:
5289:
5270:
5269:
5243:
5230:
5229:
5225:
5207:
5194:
5193:
5189:
5176:
5158:
5145:
5144:
5140:
5127:
5109:
5102:
5098:
5087:
5086:
5065:
5060:
5059:
5038:
5033:
5032:
5011:
5006:
5005:
4986:
4985:
4964:
4951:
4938:
4927:
4926:
4905:
4894:
4893:
4867:
4866:
4845:
4834:
4833:
4814:
4813:
4792:
4787:
4786:
4765:
4760:
4759:
4734:
4733:
4708:
4707:
4656:
4655:
4612:
4611:
4607:
4582:
4581:
4577:
4553:
4537:
4536:
4497:
4484:
4470:
4469:
4439:
4426:
4406:
4405:
4399:disjoint unions
4371:
4363:
4362:
4335:
4334:
4310:
4302:
4301:
4282:
4281:
4259:
4258:
4239:
4238:
4180:
4179:
4157:
4129:
4124:
4123:
4051:
4038:
4034:
4033:
4029:
3994:
3993:
3937:
3936:
3915:
3914:
3871:
3866:
3865:
3842:
3837:
3836:
3830:
3829:
3806:
3805:
3786:
3785:
3758:
3757:
3738:
3737:
3728:
3722:
3697:
3696:
3674:
3673:
3646:
3645:
3623:
3622:
3595:
3594:
3572:
3571:
3540:
3539:
3517:
3516:
3487:
3486:
3482:Hausdorff space
3459:
3458:
3439:
3438:
3391:
3390:
3352:
3351:
3348:
3316:
3311:
3310:
3287:
3282:
3281:
3260:
3259:
3230:
3229:
3225:Subsets of the
3198:
3193:
3192:
3167:
3166:
3142:
3141:
3114:
3113:
3094:
3093:
3074:
3073:
3054:
3053:
3034:
3033:
3010:
3009:
2986:
2985:
2979:
2978:
2943:
2942:
2908:
2907:
2888:
2887:
2856:
2855:
2833:
2832:
2813:
2812:
2790:
2789:
2770:
2769:
2757:
2756:
2740:
2698:
2697:
2669:
2668:
2646:
2645:
2624:
2623:
2597:
2596:
2547:
2546:
2527:
2526:
2507:
2506:
2467:
2466:
2434:
2433:
2410:
2409:
2352:
2351:
2330:
2329:
2291:
2286:
2285:
2260:
2259:
2231:
2230:
2207:
2202:
2201:
2180:
2179:
2173:Sorgenfrey line
2164:, the space of
2144:
2143:
2103:
2102:
2099:Euclidean plane
2068:
2063:
2062:
2012:
2011:
1958:
1957:
1899:
1898:
1862:
1861:
1827:
1826:
1795:
1794:
1756:
1755:
1751:
1717:
1716:
1697:
1696:
1677:
1676:
1657:
1656:
1637:
1636:
1617:
1616:
1597:
1596:
1577:
1576:
1554:
1553:
1534:
1533:
1514:
1513:
1507:
1506:
1486:
1485:
1478:
1477:
1472:
1448:
1447:
1410:
1405:
1404:
1382:
1381:
1378:quasi-component
1358:
1357:
1324:
1323:
1301:
1300:
1281:
1280:
1259:
1254:
1253:
1223:
1204:
1199:
1198:
1174:
1173:
1142:
1141:
1087:
1086:
1035:
1034:
1010:
991:
986:
985:
964:
951:
946:
945:
909:
908:
885:
884:
861:
860:
831:
830:
811:
810:
791:
790:
768:
767:
748:
747:
728:
727:
708:
707:
685:
684:
662:
661:
642:
641:
638:
630:Felix Hausdorff
606:
605:
566:
565:
534:
533:
514:
513:
512:functions from
483:
482:
461:
460:
437:
436:
414:
413:
390:
389:
367:
366:
344:
343:
309:
308:
302:
301:
281:
280:
274:
242:
241:
210:
209:
199:
198:
178:
177:
159:of two or more
149:connected space
137:
136:
135:
134:
133:
100:. Furthermore,
95:
91:
87:
83:
60:, yellow space
50:
41:
40:
28:
23:
22:
15:
12:
11:
5:
6478:
6476:
6468:
6467:
6462:
6452:
6451:
6445:
6444:
6442:
6441:
6431:
6430:
6429:
6424:
6419:
6404:
6394:
6384:
6372:
6361:
6358:
6357:
6355:
6354:
6349:
6344:
6339:
6334:
6329:
6323:
6321:
6315:
6314:
6312:
6311:
6306:
6301:
6299:Winding number
6296:
6291:
6285:
6283:
6279:
6278:
6276:
6275:
6270:
6265:
6260:
6255:
6250:
6245:
6240:
6239:
6238:
6233:
6231:homotopy group
6223:
6222:
6221:
6216:
6211:
6206:
6201:
6191:
6186:
6181:
6171:
6169:
6165:
6164:
6157:
6155:
6153:
6152:
6147:
6142:
6141:
6140:
6130:
6129:
6128:
6118:
6113:
6108:
6103:
6098:
6092:
6090:
6086:
6085:
6080:
6078:
6077:
6070:
6063:
6055:
6048:
6047:
6012:
5994:
5975:
5969:
5952:
5950:
5947:
5944:
5943:
5937:Stack Exchange
5922:
5916:Stack Exchange
5901:
5894:Charles Weibel
5886:
5879:
5858:
5851:
5833:
5819:
5800:(9): 720–726.
5783:
5782:
5780:
5777:
5776:
5775:
5770:
5764:
5756:
5750:
5744:
5738:
5733:
5723:
5722:
5706:
5703:
5691:
5690:
5683:
5675:
5659:
5639:
5619:
5616:
5588:
5585:
5582:
5562:
5542:
5539:
5538:
5537:
5534:
5531:
5524:
5521:
5514:
5507:
5504:
5501:
5494:
5487:
5484:
5481:
5478:
5475:
5459:
5456:
5453:
5450:
5430:
5410:
5407:
5404:
5401:
5398:
5378:
5358:
5342:
5339:
5328:is connected.
5317:
5297:
5277:
5256:
5250:
5246:
5242:
5237:
5233:
5228:
5224:
5220:
5214:
5210:
5206:
5201:
5197:
5192:
5188:
5183:
5179:
5175:
5171:
5165:
5161:
5157:
5152:
5148:
5143:
5139:
5134:
5130:
5126:
5122:
5116:
5112:
5108:
5105:
5101:
5097:
5094:
5072:
5068:
5045:
5041:
5018:
5014:
4993:
4971:
4967:
4963:
4958:
4954:
4950:
4945:
4941:
4937:
4934:
4912:
4908:
4904:
4901:
4887:
4874:
4852:
4848:
4844:
4841:
4821:
4799:
4795:
4772:
4768:
4747:
4744:
4741:
4721:
4718:
4715:
4704:set difference
4700:
4699:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4595:
4592:
4589:
4565:
4560:
4556:
4552:
4548:
4544:
4534:quotient space
4530:
4518:
4515:
4510:
4507:
4504:
4500:
4496:
4491:
4487:
4483:
4480:
4477:
4466:
4454:
4451:
4446:
4442:
4438:
4433:
4429:
4425:
4422:
4419:
4416:
4413:
4402:
4386:
4383:
4378:
4374:
4370:
4356:
4342:
4322:
4317:
4313:
4309:
4289:
4266:
4246:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4156:
4155:Set operations
4153:
4138:
4133:
4103:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4074:
4069:
4063:
4060:
4054:
4050:
4047:
4044:
4041:
4037:
4032:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3923:
3880:
3875:
3851:
3846:
3813:
3793:
3765:
3745:
3724:Main article:
3721:
3718:
3717:
3716:
3704:
3693:
3681:
3660:
3656:
3653:
3642:
3630:
3609:
3605:
3602:
3591:
3579:
3568:
3547:
3524:
3494:
3466:
3446:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3386:, which is an
3370:is said to be
3359:
3347:
3344:
3325:
3320:
3296:
3291:
3268:
3252:if and only if
3250:are connected
3238:
3205:
3201:
3174:
3150:
3134:path-connected
3132:is said to be
3121:
3101:
3081:
3061:
3052:equivalent to
3041:
3017:
2993:
2980:path-component
2965:
2962:
2959:
2956:
2953:
2950:
2930:
2927:
2924:
2921:
2918:
2915:
2895:
2875:
2872:
2869:
2866:
2863:
2840:
2820:
2797:
2777:
2739:
2736:
2720:
2719:
2718:
2717:
2705:
2685:
2682:
2679:
2676:
2653:
2643:
2631:
2616:
2604:
2586:
2574:
2570:
2566:
2563:
2560:
2557:
2554:
2534:
2514:
2494:
2490:
2486:
2483:
2480:
2477:
2474:
2460:
2453:
2441:
2417:
2406:
2399:
2388:
2373:
2360:
2338:
2314:
2300:
2295:
2273:
2270:
2267:
2247:
2244:
2241:
2238:
2216:
2211:
2188:
2176:
2169:
2152:
2141:
2138:
2125:
2122:
2119:
2116:
2113:
2110:
2095:
2077:
2072:
2055:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2009:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1945:
1942:
1939:
1936:
1933:
1918:
1915:
1912:
1909:
1906:
1895:
1884:
1881:
1878:
1875:
1872:
1869:
1849:
1846:
1843:
1840:
1837:
1834:
1814:
1811:
1808:
1805:
1802:
1775:
1772:
1769:
1766:
1763:
1750:
1747:
1725:
1704:
1684:
1664:
1644:
1624:
1604:
1584:
1561:
1541:
1521:
1493:
1471:
1468:
1455:
1434:
1430:
1426:
1422:
1417:
1413:
1392:
1389:
1365:
1340:
1336:
1332:
1311:
1308:
1288:
1266:
1262:
1238:
1235:
1230:
1226:
1222:
1219:
1216:
1211:
1207:
1186:
1182:
1161:
1158:
1155:
1152:
1149:
1129:
1126:
1123:
1120:
1117:
1114:
1110:
1106:
1103:
1100:
1097:
1094:
1074:
1071:
1068:
1065:
1062:
1058:
1054:
1051:
1048:
1045:
1042:
1022:
1017:
1013:
1009:
1006:
1003:
998:
994:
971:
967:
963:
958:
954:
916:
892:
868:
841:
838:
829:that contains
818:
798:
778:
775:
755:
735:
715:
692:
672:
669:
649:
637:
634:
613:
602:
601:
585:
582:
579:
576:
573:
553:
550:
547:
544:
541:
521:
506:
503:separated sets
490:
480:
468:
444:
433:
421:
397:
386:
374:
351:
327:is said to be
316:
299:is said to be
288:
273:
270:
249:
233:path connected
217:
185:
93:
89:
85:
81:
51:
44:
43:
42:
34:
33:
32:
31:
26:
24:
18:Path-connected
14:
13:
10:
9:
6:
4:
3:
2:
6477:
6466:
6463:
6461:
6458:
6457:
6455:
6440:
6432:
6428:
6425:
6423:
6420:
6418:
6415:
6414:
6413:
6405:
6403:
6399:
6395:
6393:
6389:
6385:
6383:
6378:
6373:
6371:
6363:
6362:
6359:
6353:
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6333:
6330:
6328:
6325:
6324:
6322:
6320:
6316:
6310:
6309:Orientability
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6290:
6287:
6286:
6284:
6280:
6274:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6241:
6237:
6234:
6232:
6229:
6228:
6227:
6224:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6202:
6200:
6197:
6196:
6195:
6192:
6190:
6187:
6185:
6182:
6180:
6176:
6173:
6172:
6170:
6166:
6161:
6151:
6148:
6146:
6145:Set-theoretic
6143:
6139:
6136:
6135:
6134:
6131:
6127:
6124:
6123:
6122:
6119:
6117:
6114:
6112:
6109:
6107:
6106:Combinatorial
6104:
6102:
6099:
6097:
6094:
6093:
6091:
6087:
6083:
6076:
6071:
6069:
6064:
6062:
6057:
6056:
6053:
6037:on 2016-03-04
6033:
6029:
6025:
6018:
6013:
6010:
6006:
6005:
6000:
5995:
5990:
5989:
5984:
5981:
5976:
5972:
5970:0-13-181629-2
5966:
5962:
5958:
5954:
5953:
5948:
5939:
5938:
5933:
5926:
5923:
5918:
5917:
5912:
5905:
5902:
5899:
5895:
5890:
5887:
5882:
5880:0-89874-551-9
5876:
5872:
5868:
5862:
5859:
5854:
5852:0-486-43479-6
5848:
5844:
5837:
5834:
5829:
5823:
5820:
5815:
5811:
5807:
5803:
5799:
5795:
5788:
5785:
5778:
5774:
5771:
5768:
5765:
5763:
5761:
5757:
5754:
5751:
5748:
5745:
5742:
5739:
5737:
5734:
5728:
5725:
5724:
5720:
5714:
5709:
5704:
5702:
5700:
5696:
5688:
5684:
5680:
5676:
5673:
5657:
5637:
5629:
5628:
5627:
5625:
5617:
5615:
5613:
5608:
5606:
5600:
5586:
5583:
5580:
5560:
5552:
5547:
5540:
5535:
5532:
5529:
5525:
5522:
5519:
5515:
5512:
5508:
5505:
5502:
5499:
5495:
5492:
5488:
5485:
5482:
5479:
5476:
5473:
5454:
5448:
5428:
5408:
5402:
5399:
5396:
5376:
5356:
5348:
5345:
5344:
5340:
5334:
5329:
5315:
5295:
5275:
5254:
5248:
5244:
5240:
5235:
5231:
5226:
5222:
5218:
5212:
5208:
5204:
5199:
5195:
5190:
5186:
5181:
5177:
5173:
5169:
5163:
5159:
5155:
5150:
5146:
5141:
5137:
5132:
5128:
5124:
5120:
5114:
5110:
5106:
5103:
5099:
5095:
5092:
5070:
5066:
5043:
5039:
5016:
5012:
4991:
4969:
4965:
4961:
4956:
4952:
4948:
4943:
4939:
4935:
4932:
4910:
4906:
4902:
4899:
4886:
4872:
4850:
4846:
4842:
4839:
4819:
4797:
4793:
4770:
4766:
4745:
4739:
4719:
4716:
4713:
4705:
4682:
4676:
4673:
4667:
4661:
4638:
4632:
4629:
4623:
4617:
4593:
4590:
4587:
4558:
4554:
4546:
4542:
4535:
4531:
4513:
4508:
4505:
4502:
4498:
4494:
4489:
4485:
4481:
4478:
4467:
4449:
4444:
4440:
4436:
4431:
4427:
4423:
4420:
4417:
4414:
4403:
4400:
4381:
4376:
4372:
4368:
4360:
4359:
4358:
4354:
4340:
4315:
4311:
4287:
4278:
4264:
4244:
4235:
4218:
4215:
4212:
4206:
4200:
4197:
4194:
4188:
4185:
4177:
4172:
4170:
4161:
4154:
4152:
4136:
4121:
4118:
4101:
4094:
4091:
4088:
4082:
4079:
4076:
4072:
4067:
4061:
4058:
4052:
4048:
4045:
4042:
4039:
4035:
4030:
4026:
4017:
4014:
4011:
4002:
3999:
3992:, defined as
3991:
3986:
3969:
3966:
3963:
3957:
3951:
3948:
3945:
3913:intervals in
3912:
3902:
3898:
3896:
3878:
3849:
3834:
3825:
3811:
3791:
3783:
3779:
3763:
3743:
3736:
3734:
3727:
3719:
3702:
3694:
3679:
3654:
3651:
3643:
3628:
3603:
3600:
3592:
3577:
3569:
3565:
3564:
3563:
3561:
3545:
3536:
3522:
3514:
3510:
3506:
3483:
3478:
3464:
3444:
3436:
3435:arc-component
3420:
3411:
3408:
3405:
3399:
3396:
3389:
3385:
3381:
3377:
3373:
3372:arc-connected
3357:
3345:
3343:
3341:
3323:
3294:
3257:
3253:
3228:
3223:
3221:
3203:
3199:
3191:
3186:
3172:
3164:
3139:
3135:
3119:
3099:
3079:
3059:
3039:
3031:
3015:
3007:
2991:
2983:
2963:
2960:
2954:
2948:
2928:
2925:
2919:
2913:
2893:
2870:
2867:
2864:
2854:
2853:unit interval
2838:
2818:
2811:
2795:
2775:
2768:from a point
2767:
2766:
2761:
2749:
2744:
2737:
2735:
2733:
2729:
2725:
2703:
2683:
2680:
2677:
2674:
2667:
2651:
2644:
2629:
2621:
2617:
2602:
2594:
2593:
2591:
2587:
2564:
2561:
2555:
2552:
2532:
2512:
2484:
2481:
2475:
2472:
2465:
2461:
2458:
2454:
2439:
2431:
2415:
2407:
2404:
2400:
2397:
2393:
2389:
2386:
2382:
2378:
2374:
2327:
2323:
2322:Hilbert space
2319:
2315:
2298:
2271:
2268:
2265:
2245:
2242:
2239:
2236:
2214:
2177:
2174:
2170:
2167:
2142:
2139:
2123:
2117:
2114:
2111:
2100:
2096:
2093:
2075:
2060:
2059:convex subset
2056:
2038:
2032:
2026:
2023:
2020:
2010:
1996:
1990:
1987:
1984:
1978:
1972:
1969:
1966:
1940:
1937:
1934:
1913:
1910:
1907:
1897:The union of
1896:
1882:
1876:
1873:
1870:
1847:
1841:
1838:
1835:
1809:
1806:
1803:
1792:
1789:
1770:
1767:
1764:
1753:
1752:
1748:
1746:
1744:
1740:
1702:
1682:
1662:
1642:
1622:
1602:
1582:
1575:
1559:
1539:
1519:
1511:
1491:
1483:
1482:
1469:
1467:
1453:
1432:
1428:
1420:
1415:
1390:
1387:
1379:
1363:
1355:
1338:
1334:
1309:
1306:
1286:
1264:
1250:
1236:
1233:
1228:
1224:
1220:
1217:
1214:
1209:
1205:
1184:
1156:
1153:
1150:
1127:
1121:
1118:
1115:
1112:
1104:
1101:
1095:
1092:
1069:
1066:
1063:
1060:
1052:
1049:
1043:
1040:
1033:and then set
1020:
1015:
1011:
1007:
1004:
1001:
996:
992:
969:
965:
961:
956:
952:
943:
939:
934:
933:closed subset
930:
914:
906:
890:
882:
866:
859:
855:
839:
836:
816:
796:
776:
773:
766:that contain
753:
733:
713:
706:
690:
670:
667:
647:
635:
633:
631:
627:
626:Frigyes Riesz
611:
599:
580:
577:
574:
548:
545:
542:
519:
511:
507:
504:
488:
481:
466:
458:
442:
434:
419:
411:
395:
387:
372:
365:
364:
363:
349:
340:
338:
334:
330:
314:
306:
286:
279:
271:
269:
267:
266:
261:
247:
238:
234:
229:
215:
207:
203:
200:connected set
183:
174:
172:
168:
165:
162:
158:
154:
150:
146:
142:
131:
127:
123:
119:
115:
111:
107:
103:
99:
79:
75:
71:
67:
63:
59:
56:, pink space
55:
48:
38:
30:
19:
6439:Publications
6304:Chern number
6294:Betti number
6203:
6177: /
6168:Key concepts
6116:Differential
6039:. Retrieved
6032:the original
6027:
6023:
6002:
5986:
5960:
5935:
5925:
5914:
5904:
5889:
5870:
5861:
5842:
5836:
5822:
5797:
5793:
5787:
5759:
5692:
5621:
5609:
5601:
5573:-cycle with
5544:
5346:
4891:
4701:
4279:
4236:
4175:
4173:
4169:intersection
4168:
4166:
3987:
3907:
3828:
3826:
3780:if it has a
3777:
3731:
3729:
3567:cardinality.
3537:
3479:
3434:
3375:
3371:
3349:
3258:and rays of
3224:
3187:
3162:
3137:
3133:
3112:. The space
3032:which makes
2977:
2763:
2755:
2753:
2747:
2721:
2615:is connected
2326:Banach space
2166:real numbers
1505:
1476:
1473:
1251:
880:
704:
639:
603:
341:
328:
303:disconnected
300:
275:
263:
230:
197:
175:
167:open subsets
148:
138:
132:has genus 4.
129:
125:
121:
117:
105:
101:
98:disconnected
97:
73:
69:
65:
61:
57:
53:
36:
29:
6402:Wikiversity
6319:Key results
5031:, and thus
4115:, with the
3378:if any two
2788:to a point
2408:If a space
2320:, e.g. any
1635:containing
1595:containing
927:: they are
455:with empty
410:clopen sets
145:mathematics
6454:Categories
6248:CW complex
6189:Continuity
6179:Closed set
6138:cohomology
6041:2010-05-17
5779:References
5762:-connected
5682:connected.
4984:. Because
3935:, such as
3735:at a point
3163:-connected
3028:under the
2666:idempotent
2590:local ring
2403:Cantor set
2137:connected.
1655:such that
1504:is called
942:singletons
510:continuous
260:-connected
116:0), while
6427:geometric
6422:algebraic
6273:Cobordism
6209:Hausdorff
6204:connected
6121:Geometric
6111:Continuum
6101:Algebraic
6009:EMS Press
5988:MathWorld
5406:→
5241:∩
5223:∪
5205:∪
5174:∪
5156:∪
5125:∪
5107:∪
4962:∪
4936:∪
4903:∪
4843:∪
4743:∖
4717:⊇
4630:∪
4591:∪
4517:∅
4514:≠
4495:∩
4476:∀
4453:∅
4450:≠
4437:∩
4412:∀
4385:∅
4382:≠
4369:⋂
4207:∪
4083:∈
4049:
4027:∪
3958:∪
3911:separated
3824:is open.
3655:×
3604:×
3493:Δ
3418:→
3388:embedding
3256:intervals
3227:real line
3204:∗
3190:long line
2851:from the
2675:≠
2556:
2476:
2269:≥
2240:≥
2033:∪
1979:∪
1743:Hausdorff
1574:open sets
1425:Γ
1421:⊂
1412:Γ
1331:Γ
1261:Γ
1234:∈
1215:∈
1105:∈
1053:∈
905:partition
867:⊆
858:inclusion
797:⊆
337:empty set
329:connected
164:non-empty
124:are not:
108:are also
76:(made of
6392:Wikibook
6370:Category
6258:Manifold
6226:Homotopy
6184:Interior
6175:Open set
6133:Homology
6082:Topology
5959:(2000).
5869:(1968).
5705:See also
5677:Since a
5528:manifold
5511:quotient
5341:Theorems
3350:A space
3218:and the
2229:, where
1788:standard
1749:Examples
1433:′
1376:(called
1339:′
929:disjoint
907:of
457:boundary
206:subspace
161:disjoint
141:topology
68:are all
6417:general
6219:uniform
6199:compact
6150:Digital
5814:2321676
5551:5-cycle
5518:product
5516:Every
5491:closure
4120:induced
3641:is not.
3558:be the
2724:annulus
2696:(i.e.,
2664:has no
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1403:) Then
903:form a
854:maximal
92:, and E
78:subsets
6412:Topics
6214:metric
6089:Fields
5967:
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5812:
5546:Graphs
5541:Graphs
5526:Every
5509:Every
5498:closed
5349:: Let
3480:Every
3004:is an
2618:Every
2375:Every
1354:clopen
628:, and
412:) are
333:subset
239:, and
6194:Space
6035:(PDF)
6020:(PDF)
5810:JSTOR
4889:Proof
4610:then
4176:union
3433:. An
2906:with
2808:in a
2728:disks
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331:. A
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5965:ISBN
5875:ISBN
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