2868:
By itself, Definition 3 is not quite as useful as the other two definitions as it lacks some of the properties implied by the others. For example, every quotient space of a space satisfying
Definition 1 or Definition 2 is a space of the same kind. But that does not hold for Definition 3.
2984:
section, there is no universally accepted definition in the literature for compactly generated spaces; but
Definitions 1, 2, 3 from that section are some of the more commonly used. In order to express results in a more concise way, this section will make use of the abbreviations
4148:
on a set induced by a family of functions from CG-1 spaces is also CG-1. And the same holds for CG-2. This follows by combining the results above for disjoint unions and quotient spaces, together with the behavior of final topologies under composition of functions.
69:
in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like
4072:
A quotient space of a CG-3 space is not CG-3 in general. In fact, every CG-2 space is a quotient space of a CG-3 space (namely, some locally compact
Hausdorff space); but there are CG-2 spaces that are not CG-3. For a concrete example, the
3701:, which is not compactly generated (as mentioned in the Examples section, it is anticompact and non-discrete). Another example is the Arens space, which is sequential Hausdorff, hence compactly generated. It contains as a subspace the
2923:
need not be a CW-complex. By contrast, the category of simplicial sets had many convenient properties, including being cartesian closed. The history of the study of repairing this situation is given in the article on the
1620:
For coherent spaces, that corresponds to showing that the space is coherent with a subfamily of the family of subspaces. For example, this provides one way to show that locally compact spaces are compactly generated.
4244:
3214:
Compactly generated
Hausdorff spaces include the Hausdorff version of the various classes of spaces mentioned above as CG-1 or CG-2, namely Hausdorff sequential spaces, Hausdorff first countable spaces,
3708:
In a CG-1 space, every closed set is CG-1. The same does not hold for open sets. For instance, as shown in the
Examples section, there are many spaces that are not CG-1, but they are open in their
1571:
5846:
4312:
2737:
1412:
3930:
2969:
2877:
2480:
4891:
4640:
5889:
5649:
5304:
3118:
5745:
5683:
5582:
3976:
961:
512:
5205:
4752:
1618:
861:
477:
5711:
5611:
3637:
1537:
1236:
1189:
1065:
1041:
607:
556:
427:
4920:
4669:
2649:
1828:
234:
116:
4444:
3814:
3449:
3399:
3370:
3144:
2554:
2262:
1981:
The equivalence between conditions (1) and (2) follows from the fact that every inclusion from a subspace is a continuous map; and on the other hand, every continuous map
1733:
1313:
743:
5331:
3695:
3664:
4807:
4405:
4342:
3873:
2693:
2396:
2321:
782:
5925:
5354:
4484:
3768:
2210:
2011:
1888:
834:
2997:
to denote each of the three definitions unambiguously. This is summarized in the table below (see the
Definitions section for other equivalent conditions for each).
2600:
1779:
1359:
185:
5108:
5005:
2864:
642:
5952:
5498:
4139:
4003:
2789:
2763:
1163:
342:
2089:
2060:
5380:
987:
5788:
5521:
5446:
5403:
5228:
4970:
4061:
3736:
3496:
3167:
2233:
2112:
1911:
1435:
1259:
1212:
1131:
1108:
912:
717:
583:
450:
393:
5996:
5976:
5765:
5541:
5466:
5423:
5268:
5248:
5164:
5140:
5069:
5045:
5025:
4947:
4849:
4829:
4772:
4711:
4691:
4595:
4575:
4552:
4532:
4504:
4034:
3834:
3788:
3556:
3536:
3516:
3469:
3427:
3300:
3280:
3256:
3187:
2620:
2574:
2512:
2447:
2420:
2361:
2341:
2282:
2174:
2142:
2031:
1964:
1933:
1852:
1799:
1753:
1700:
1668:
1591:
1513:
1482:
1458:
1379:
1333:
1279:
1085:
1009:
932:
889:
802:
686:
662:
532:
366:
274:
254:
205:
159:
135:
55:
4107:
2829:
1647:
Informally, a space whose topology is determined by its compact subspaces, or equivalently in this case, by all continuous maps from arbitrary compact spaces.
6851:
2461:
article, condition (1) is well-defined, even though the family of continuous maps from arbitrary compact
Hausdorff spaces is not a set but a proper class.
4318:
from the real line with the positive integers identified to a point is sequential. Both spaces are compactly generated
Hausdorff, but their product
3197:. So among the finite spaces, which are all CG-2, the CG-3 spaces are the ones with the discrete topology. Any finite non-discrete space, like the
6622:
3049:
Every CG-3 space is CG-2 and every CG-2 space is CG-1. The converse implications do not hold in general, as shown by some of the examples below.
1978:
article, condition (2) is well-defined, even though the family of continuous maps from arbitrary compact spaces is not a set but a proper class.
6005:
These ideas can be generalized to the non-Hausdorff case. This is useful since identification spaces of
Hausdorff spaces need not be Hausdorff.
6797:
6748:
6692:
4156:
of CG-1 spaces is CG-1. The same holds for CG-2. This is also an application of the results above for disjoint unions and quotient spaces.
2880:
can also be defined by pairing the weak
Hausdorff property with Definition 3, which may be easier to state and work with than Definition 2.
4506:
is compactly generated according to one of the definitions in this article. Since compactly generated spaces are defined in terms of a
2953:
These ideas generalize to the non-Hausdorff case; i.e. with a different definition of compactly generated spaces. This is useful since
2950:
is given below. Another suggestion (1964) was to consider the usual Hausdorff spaces but use functions continuous on compact subsets.
6662:
6635:
6596:
6575:
4175:
6821:
2900:, and can be found in General Topology by Kelley, Topology by Dugundji, Rational Homotopy Theory by Félix, Halperin, and Thomas.
3578:
Subspaces of a compactly generated space are not compactly generated in general, even in the Hausdorff case. For example, the
1492:) as part of the definition, while others don't. The definitions in this article will not comprise any such separation axiom.
5585:
2904:
288:
4362:
is in the sense of condition (3) in the corresponding article, namely each point has a local base of compact neighborhoods.)
1542:
6852:
https://math.stackexchange.com/questions/4646084/unraveling-the-various-definitions-of-k-space-or-compactly-generated-space
4454:
The continuous functions on compactly generated spaces are those that behave well on compact subsets. More precisely, let
6037:
5800:
6561:
6731:
4257:
3709:
3472:
2465:
6866:
6019:
4598:
4366:
4066:
4037:
4009:
3891:
3667:
3216:
3067:
2908:
2450:
2121:
Informally, a space whose topology is determined by all continuous maps from arbitrary compact Hausdorff spaces.
1936:
292:
280:
with respect to this family of maps. And other variations of the definition replace compact spaces with compact
6871:
6028:
2943:
2654:
Every space satisfying Definition 3 also satisfies Definition 2. The converse is not true. For example, the
2464:
Every space satisfying Definition 2 also satisfies Definition 1. The converse is not true. For example, the
6348:
4714:
2698:
2475:
Definition 2 is the one more commonly used in algebraic topology. This definition is often paired with the
1214:
for example all compact subspaces, or all compact Hausdorff subspaces. This corresponds to choosing a set
6839:
6726:
6680:
4374:
4013:
3208:
1967:
1384:
405:
in the literature. These definitions share a common structure, starting with a suitably specified family
4168:
of two compactly generated spaces need not be compactly generated, even if both spaces are Hausdorff and
4069:
space is CG-2. Conversely, every CG-2 space is the quotient space of a locally compact Hausdorff space.
3896:
3875:
is also a quotient map on a locally compact Hausdorff space). The same is true more generally for every
2932:
1624:
Below are some of the more commonly used definitions in more detail, in increasing order of specificity.
1414:
Another choice is to take the family of all continuous maps from arbitrary spaces of a certain type into
6014:
5999:
5048:
4854:
4603:
4511:
3063:
3026:
Topology same as final topology with respect to continuous maps from arbitrary compact Hausdorff spaces
5857:
5617:
5276:
3088:
6827:
6146:
5716:
5654:
5553:
4554:
with the various maps in the family used to define the final topology. The specifics are as follows.
3935:
2942:
of compactly generated Hausdorff spaces, which is in fact cartesian closed. These ideas extend on the
937:
488:
5177:
4719:
3311:
3224:
2954:
1596:
1437:
for example all such maps from arbitrary compact spaces, or from arbitrary compact Hausdorff spaces.
839:
455:
75:
5688:
5588:
3584:
1518:
1217:
1170:
1046:
1022:
588:
537:
408:
6711:
4896:
4645:
4408:
4074:
3198:
2655:
2625:
2523:
1804:
1703:
1286:
210:
92:
17:
4414:
3793:
3432:
3382:
3353:
3123:
2903:
The motivation for their deeper study came in the 1960s from well known deficiencies of the usual
2533:
2241:
1712:
1635:
is unambiguous and refers to a compactly generated space (in any of the definitions) that is also
1292:
722:
6789:
6736:
6684:
5851:
5309:
4165:
3673:
3642:
3518:
is compact, hence CG-1. But it is not CG-2 because open subspaces inherit the CG-2 property and
3373:
3347:
2961:
2947:
2916:
296:
3042:
For Hausdorff spaces the properties CG-1, CG-2, CG-3 are equivalent. Such spaces can be called
6824:- contains an excellent catalog of properties and constructions with compactly generated spaces
6649:
4777:
4384:
4321:
3839:
2660:
2366:
2287:
748:
6803:
6793:
6744:
6722:
6698:
6688:
6658:
6631:
6617:
6592:
6584:
6571:
5894:
5336:
4457:
4315:
4247:
3741:
3670:
is compact Hausdorff, hence compactly generated. Its subspace with all limit ordinals except
3402:
3194:
2912:
2792:
2183:
1984:
1861:
1495:
As an additional general note, a sufficient condition that can be useful to show that a space
807:
369:
345:
295:
while still containing the typical spaces of interest, which makes them convenient for use in
86:
35:
4347:
However, in some cases the product of two compactly generated spaces is compactly generated:
2579:
1758:
1338:
891:
are the closed sets in its k-ification, with a corresponding characterization. In the space
164:
6279:
6236:
6078:
5791:
5078:
4975:
4378:
4169:
3702:
3698:
3325:
3318:
3059:
2939:
2834:
2469:
612:
6758:
5930:
5471:
4112:
3981:
2768:
2742:
1136:
315:
6767:
6754:
6740:
4355:
4251:
3429:
that is not CG-1 (for example the Arens-Fort space or an uncountable product of copies of
2739:
does not satisfy Definition 3, because its compact Hausdorff subspaces are the singletons
2423:
2065:
2036:
1940:
1636:
1485:
281:
138:
71:
5359:
966:
5770:
5503:
5428:
5385:
5210:
4952:
4043:
3718:
3478:
3234:
To provide examples of spaces that are not compactly generated, it is useful to examine
3149:
2215:
2094:
1893:
1417:
1241:
1194:
1113:
1090:
894:
699:
565:
432:
375:
5981:
5961:
5955:
5750:
5526:
5451:
5408:
5271:
5253:
5233:
5149:
5143:
5125:
5072:
5054:
5030:
5010:
4932:
4834:
4814:
4757:
4696:
4676:
4580:
4560:
4537:
4517:
4507:
4489:
4145:
4019:
3876:
3819:
3773:
3541:
3521:
3501:
3454:
3412:
3285:
3265:
3241:
3172:
3053:
2965:
2873:
2605:
2559:
2526:
with the family of its compact Hausdorff subspaces; namely, it satisfies the property:
2497:
2476:
2458:
2432:
2405:
2346:
2326:
2267:
2177:
2159:
2127:
2016:
1975:
1949:
1918:
1855:
1837:
1784:
1738:
1685:
1653:
1576:
1498:
1489:
1467:
1443:
1364:
1318:
1264:
1070:
994:
917:
874:
787:
671:
665:
647:
517:
483:
351:
277:
259:
239:
190:
144:
120:
40:
6022: – topological space in which the topology is determined by its countable subsets
5425:
coincide, and the induced topologies on compact subsets are the same. It follows that
4080:
2802:
6860:
6785:
6672:
6221:
3579:
3204:
2938:
The first suggestion (1962) to remedy this situation was to restrict oneself to the
2491:
Informally, a space whose topology is determined by its compact Hausdorff subspaces.
2472:
is compact and hence satisfies Definition 1, but it does not satisfies Definition 2.
66:
4065:
A quotient space of a CG-2 space is CG-2. In particular, every quotient space of a
3738:
every closed set is CG-2; and so is every open set (because there is a quotient map
287:
Compactly generated spaces were developed to remedy some of the shortcomings of the
6645:
6222:"Monoidal closed, Cartesian closed and convenient categories of topological spaces"
4377:
of compactly generated spaces (like all CG-1 spaces or all CG-2 spaces), the usual
4036:
is the quotient space of a weakly locally compact space, which can be taken as the
3332:
3220:
3071:
2796:
6779:
6565:
6181:
5795:
3409:
For examples of spaces that are CG-1 and not CG-2, one can start with any space
1627:
For Hausdorff spaces, all three definitions are equivalent. So the terminology
3342:
Other examples of (Hausdorff) spaces that are not compactly generated include:
5547:
3228:
2920:
6284:
6267:
6031: – topology in which the intersection of any family of open sets is open
1706:
with the family of its compact subspaces; namely, it satisfies the property:
6702:
4153:
3317:
The one-point Lindelöfication of an uncountable discrete space (also called
3238:
spaces, that is, spaces whose compact subspaces are all finite. If a space
6240:
6807:
1087:
is equal to its k-ification; equivalently, if every k-open set is open in
5111:
4486:
be a function from a topological space to another and suppose the domain
4351:
The product of two first countable spaces is first countable, hence CG-2.
4077:
is not CG-3, but is homeomorphic to the quotient of the compact interval
3078:
2897:
236:
Other definitions use a family of continuous maps from compact spaces to
31:
6082:
914:
every open set is k-open and every closed set is k-closed. The space
4446:
does belong to the expected category and is the categorical product.
2795:
instead. On the other hand, it satisfies Definition 2 because it is
89:
with the family of its compact subspaces, meaning that for every set
6163:
6161:
4005:
is CG-1. The corresponding statements also hold for CG-2 and CG-3.
4012:
of a CG-1 space is CG-1. In particular, every quotient space of a
3036:
Topology coherent with family of its compact Hausdorff subspaces
1191:, one can take all the inclusions maps from certain subspaces of
6843:
6831:
6150:
3879:
set, that is, the intersection of an open set and a closed set.
3570:
section for the meaning of the abbreviations CG-1, CG-2, CG-3.)
3211:
spaces are CG-1, but not necessarily CG-2 (see examples below).
2925:
6607:
6203:
6201:
3282:
has the discrete topology and the corresponding k-ification of
2062:
and thus factors through the inclusion of the compact subspace
1440:
These different choices for the family of continuous maps into
2919:
is not always an identification map, and the usual product of
2876:
spaces Definitions 2 and 3 are equivalent. Thus the category
6396:
6394:
6069:
Lawson, J.; Madison, B. (1974). "Quotients of k-semigroups".
4239:{\displaystyle X=\mathbb {R} \setminus \{1,1/2,1/3,\ldots \}}
452:
The various definitions differ in their choice of the family
291:. In particular, under some of the definitions, they form a
276:
to be compactly generated if its topology coincides with the
4407:
is not compactly generated in general, so cannot serve as a
2152:
if it satisfies any of the following equivalent conditions:
1678:
if it satisfies any of the following equivalent conditions:
1602:
1558:
1548:
1524:
1396:
1223:
1176:
1052:
1028:
947:
845:
594:
543:
498:
461:
414:
78:) in the definition of one or both terms, and others don't.
2970:
category CGWH of compactly generated weak Hausdorff spaces
2481:
category CGWH of compactly generated weak Hausdorff spaces
3978:
of topological spaces is CG-1 if and only if each space
6612:, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk
3302:
is the discrete topology. Therefore, any anticompact T
3146:
is the empty set or a single point, which is closed in
3120:
its intersection with every compact Hausdorff subspace
3016:
Topology coherent with family of its compact subspaces
6118:
6116:
1566:{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}
5984:
5964:
5933:
5897:
5860:
5803:
5773:
5753:
5719:
5691:
5657:
5620:
5591:
5556:
5529:
5506:
5474:
5454:
5431:
5411:
5388:
5362:
5339:
5312:
5279:
5256:
5236:
5213:
5180:
5152:
5128:
5081:
5057:
5033:
5013:
4978:
4955:
4935:
4899:
4857:
4837:
4817:
4780:
4760:
4722:
4699:
4679:
4648:
4606:
4583:
4563:
4540:
4520:
4492:
4460:
4417:
4387:
4324:
4260:
4178:
4115:
4083:
4046:
4022:
3984:
3938:
3899:
3842:
3822:
3796:
3776:
3744:
3721:
3676:
3645:
3587:
3544:
3524:
3504:
3481:
3457:
3435:
3415:
3385:
3356:
3288:
3268:
3244:
3175:
3152:
3126:
3091:
2837:
2805:
2791:, and the coherent topology they induce would be the
2771:
2745:
2701:
2663:
2628:
2608:
2582:
2562:
2536:
2500:
2435:
2408:
2369:
2349:
2329:
2290:
2270:
2244:
2218:
2186:
2162:
2130:
2097:
2068:
2039:
2019:
1987:
1952:
1921:
1896:
1864:
1840:
1807:
1787:
1761:
1741:
1715:
1688:
1656:
1599:
1579:
1545:
1521:
1501:
1470:
1446:
1420:
1387:
1367:
1341:
1321:
1295:
1267:
1244:
1220:
1197:
1173:
1139:
1116:
1093:
1073:
1049:
1025:
997:
969:
940:
920:
897:
877:
842:
810:
790:
751:
725:
702:
674:
650:
615:
591:
568:
540:
520:
491:
458:
435:
411:
378:
354:
318:
262:
242:
213:
193:
167:
147:
123:
95:
43:
6033:
Pages displaying wikidata descriptions as a fallback
6024:
Pages displaying wikidata descriptions as a fallback
5685:
with objects the Hausdorff spaces. The functor from
3056:
spaces the properties CG-2 and CG-3 are equivalent.
5841:{\displaystyle \mathbf {CGTop} \to \mathbf {Top} .}
3306:non-discrete space is not CG-1. Examples include:
688:. The open sets in the k-ification are called the
397:There are multiple (non-equivalent) definitions of
5990:
5970:
5946:
5919:
5883:
5840:
5782:
5759:
5739:
5705:
5677:
5643:
5605:
5576:
5535:
5515:
5492:
5460:
5440:
5417:
5397:
5374:
5348:
5325:
5298:
5262:
5242:
5222:
5199:
5166:that is compactly generated, sometimes called the
5158:
5134:
5102:
5063:
5039:
5019:
4999:
4964:
4941:
4914:
4885:
4843:
4823:
4801:
4766:
4746:
4705:
4685:
4663:
4634:
4589:
4569:
4546:
4526:
4498:
4478:
4438:
4399:
4336:
4306:
4238:
4133:
4101:
4055:
4028:
3997:
3970:
3924:
3867:
3828:
3808:
3782:
3762:
3730:
3689:
3658:
3631:
3550:
3530:
3510:
3490:
3463:
3443:
3421:
3393:
3364:
3294:
3274:
3250:
3181:
3161:
3138:
3112:
2964:, this property is most commonly coupled with the
2888:Compactly generated spaces were originally called
2858:
2823:
2783:
2757:
2731:
2687:
2643:
2614:
2594:
2568:
2548:
2506:
2441:
2414:
2390:
2355:
2335:
2315:
2276:
2256:
2227:
2204:
2180:with respect to the family of all continuous maps
2168:
2136:
2106:
2083:
2054:
2025:
2005:
1958:
1927:
1905:
1882:
1858:with respect to the family of all continuous maps
1846:
1822:
1793:
1773:
1747:
1727:
1694:
1662:
1612:
1585:
1565:
1531:
1507:
1476:
1452:
1429:
1406:
1373:
1353:
1327:
1307:
1273:
1253:
1230:
1206:
1183:
1157:
1125:
1102:
1079:
1059:
1035:
1003:
981:
955:
926:
906:
883:
855:
828:
796:
776:
737:
711:
680:
656:
636:
601:
577:
550:
526:
506:
471:
444:
421:
387:
360:
336:
268:
248:
228:
199:
179:
153:
129:
110:
49:
5614:with objects the compactly generated spaces, and
4534:in terms of the continuity of the composition of
4307:{\displaystyle Y=\mathbb {R} /\{1,2,3,\ldots \}}
3201:, is an example of CG-2 space that is not CG-3.
4754:is continuous for each compact Hausdorff space
1043:) if its topology is determined by all maps in
429:of continuous maps from some compact spaces to
6268:"The total negation of a topological property"
2799:to the quotient space of the compact interval
372:, that is, the collection of all open sets in
6739:reprint of 1978 ed.). Berlin, New York:
6460:
6400:
5007:denote the space of all continuous maps from
4851:is continuous if and only if the restriction
4016:space is CG-1. Conversely, every CG-1 space
1289:with that family of subspaces; namely, a set
8:
6298:
5194:
5181:
4862:
4611:
4301:
4277:
4233:
4193:
3098:
3092:
2778:
2772:
2752:
2746:
2726:
2717:
2711:
2702:
2682:
2670:
2235:In other words, it satisfies the condition:
1167:As for the different choices for the family
5468:was compactly generated to start with then
3882:In a CG-3 space, every closed set is CG-3.
2957:of Hausdorff spaces need not be Hausdorff.
6532:
6472:
6373:
6253:
6167:
6134:
6056:
3379:The product of uncountably many copies of
2831:obtained by identifying all the points in
6840:Convenient category of topological spaces
6448:
6334:
6283:
5983:
5963:
5938:
5932:
5908:
5896:
5861:
5859:
5824:
5804:
5802:
5772:
5752:
5720:
5718:
5692:
5690:
5658:
5656:
5621:
5619:
5592:
5590:
5557:
5555:
5528:
5505:
5473:
5453:
5430:
5410:
5387:
5382:One can show that the compact subsets of
5361:
5338:
5317:
5311:
5290:
5278:
5255:
5235:
5212:
5188:
5179:
5151:
5127:
5080:
5056:
5032:
5012:
4977:
4954:
4934:
4898:
4893:is continuous for each compact Hausdorff
4865:
4856:
4836:
4816:
4779:
4759:
4721:
4698:
4678:
4647:
4614:
4605:
4582:
4562:
4539:
4519:
4491:
4459:
4416:
4386:
4323:
4272:
4268:
4267:
4259:
4219:
4205:
4186:
4185:
4177:
4114:
4082:
4045:
4021:
3989:
3983:
3956:
3946:
3937:
3916:
3906:
3901:
3898:
3847:
3841:
3821:
3795:
3775:
3770:for some locally compact Hausdorff space
3743:
3720:
3681:
3675:
3650:
3644:
3620:
3592:
3586:
3543:
3523:
3503:
3480:
3456:
3437:
3436:
3434:
3414:
3387:
3386:
3384:
3358:
3357:
3355:
3287:
3267:
3243:
3174:
3151:
3125:
3090:
2836:
2804:
2770:
2744:
2700:
2662:
2627:
2607:
2581:
2561:
2535:
2499:
2434:
2407:
2368:
2348:
2328:
2295:
2289:
2269:
2243:
2217:
2185:
2161:
2129:
2096:
2067:
2038:
2018:
1986:
1951:
1920:
1895:
1863:
1839:
1806:
1786:
1760:
1740:
1714:
1687:
1655:
1601:
1600:
1598:
1578:
1557:
1556:
1547:
1546:
1544:
1523:
1522:
1520:
1500:
1469:
1445:
1419:
1395:
1394:
1386:
1366:
1340:
1320:
1294:
1266:
1243:
1222:
1221:
1219:
1196:
1175:
1174:
1172:
1138:
1115:
1092:
1072:
1051:
1050:
1048:
1027:
1026:
1024:
996:
968:
946:
945:
939:
919:
896:
876:
844:
843:
841:
809:
789:
756:
750:
724:
701:
673:
668:than (or equal to) the original topology
649:
614:
593:
592:
590:
567:
542:
541:
539:
519:
497:
496:
490:
460:
459:
457:
434:
413:
412:
410:
377:
353:
317:
261:
241:
212:
192:
166:
146:
122:
94:
42:
6657:. Chicago: University of Chicago Press.
6623:Categories for the Working Mathematician
6496:
6436:
6361:
6322:
5207:denote the family of compact subsets of
3231:are also Hausdorff compactly generated.
2999:
1515:is compactly generated (with respect to
6544:
6520:
6310:
6107:
6095:
6049:
4190:
1593:is compactly generated with respect to
6651:A Concise Course in Algebraic Topology
2732:{\displaystyle \{\emptyset ,\{1\},X\}}
1464:. Additionally, some authors require
1110:or if every k-closed set is closed in
6828:Compactly generated topological space
6424:
6412:
6207:
6147:compactly generated topological space
6122:
2622:for every compact Hausdorff subspace
308:General framework for the definitions
7:
6609:On the foundations of k-group theory
6508:
6484:
6385:
6220:Booth, Peter; Tillotson, J. (1980).
3705:, which is not compactly generated.
1484:to satisfy a separation axiom (like
1407:{\displaystyle K\in {\mathcal {C}}.}
1067:, in the sense that the topology on
3925:{\displaystyle {\coprod }_{i}X_{i}}
2968:property, so that one works in the
1631:compactly generated Hausdorff space
18:Compactly generated Hausdorff space
5543:(i.e., there are more open sets).
4886:{\displaystyle f\vert _{K}:K\to Y}
4635:{\displaystyle f\vert _{K}:K\to Y}
4365:The product of a CG-2 space and a
4354:The product of a CG-1 space and a
3338:The "Single ultrafilter topology".
2705:
2343:for every compact Hausdorff space
2212:from all compact Hausdorff spaces
25:
6630:(2nd ed.). Springer-Verlag.
5884:{\displaystyle \mathbf {CGHaus} }
5644:{\displaystyle \mathbf {CGHaus} }
5299:{\displaystyle A\cap K_{\alpha }}
4713:is continuous if and only if the
4597:is continuous if and only if the
3169:hence the singleton is closed in
3113:{\displaystyle \{x\}\subseteq X,}
3031:
3021:
3011:
1460:lead to different definitions of
65:if its topology is determined by
6626:. Graduate Texts in Mathematics
5877:
5874:
5871:
5868:
5865:
5862:
5831:
5828:
5825:
5817:
5814:
5811:
5808:
5805:
5740:{\displaystyle \mathbf {CGTop} }
5733:
5730:
5727:
5724:
5721:
5699:
5696:
5693:
5678:{\displaystyle \mathbf {CGTop} }
5671:
5668:
5665:
5662:
5659:
5637:
5634:
5631:
5628:
5625:
5622:
5599:
5596:
5593:
5577:{\displaystyle \mathbf {CGTop} }
5570:
5567:
5564:
5561:
5558:
3971:{\displaystyle (X_{i})_{i\in I}}
2981:
956:{\displaystyle T_{\mathcal {F}}}
507:{\displaystyle T_{\mathcal {F}}}
6349:"A note about the Arens' space"
6272:Illinois Journal of Mathematics
5200:{\displaystyle \{K_{\alpha }\}}
4747:{\displaystyle f\circ u:K\to Y}
4642:is continuous for each compact
2933:convenient categories of spaces
1613:{\displaystyle {\mathcal {G}}.}
934:together with the new topology
856:{\displaystyle {\mathcal {F}}.}
472:{\displaystyle {\mathcal {F}},}
6229:Pacific Journal of Mathematics
5914:
5901:
5821:
5706:{\displaystyle \mathbf {Top} }
5606:{\displaystyle \mathbf {Top} }
5230:We define the new topology on
5097:
5085:
4994:
4982:
4877:
4790:
4738:
4626:
4470:
4433:
4421:
4128:
4116:
4096:
4084:
3953:
3939:
3862:
3856:
3754:
3632:{\displaystyle \omega _{1}+1=}
3626:
3607:
3350:of uncountably many copies of
2905:category of topological spaces
2850:
2838:
2818:
2806:
2576:exactly when the intersection
2379:
2310:
2304:
2196:
2078:
2072:
2049:
2043:
1997:
1874:
1755:exactly when the intersection
1532:{\displaystyle {\mathcal {F}}}
1335:exactly when the intersection
1231:{\displaystyle {\mathcal {C}}}
1184:{\displaystyle {\mathcal {F}}}
1060:{\displaystyle {\mathcal {F}}}
1036:{\displaystyle {\mathcal {F}}}
820:
771:
765:
628:
616:
602:{\displaystyle {\mathcal {F}}}
551:{\displaystyle {\mathcal {F}}}
422:{\displaystyle {\mathcal {F}}}
331:
319:
289:category of topological spaces
81:In the simplest definition, a
27:Property of topological spaces
1:
6768:"The category of CGWH spaces"
6591:. Heldermann Verlag, Berlin.
6038:K-space (functional analysis)
4915:{\displaystyle K\subseteq X.}
4664:{\displaystyle K\subseteq X.}
3697:removed is isomorphic to the
3567:
3219:spaces, etc. In particular,
3044:compactly generated Hausdorff
2644:{\displaystyle K\subseteq X.}
1823:{\displaystyle K\subseteq X.}
1285:exactly when its topology is
229:{\displaystyle K\subseteq X.}
111:{\displaystyle A\subseteq X,}
6766:Strickland, Neil P. (2009).
6712:"Compactly generated spaces"
5122:Given any topological space
4439:{\displaystyle k(X\times Y)}
4344:is not compactly generated.
4040:of the compact subspaces of
3809:{\displaystyle U\subseteq X}
3444:{\displaystyle \mathbb {R} }
3394:{\displaystyle \mathbb {Z} }
3365:{\displaystyle \mathbb {R} }
3262:, every compact subspace of
3139:{\displaystyle K\subseteq X}
2549:{\displaystyle A\subseteq X}
2426:of compact Hausdorff spaces.
2257:{\displaystyle A\subseteq X}
1728:{\displaystyle A\subseteq X}
1308:{\displaystyle A\subseteq X}
1019:(with respect to the family
738:{\displaystyle U\subseteq X}
6732:Counterexamples in Topology
5448:is compactly generated. If
5326:{\displaystyle K_{\alpha }}
3690:{\displaystyle \omega _{1}}
3659:{\displaystyle \omega _{1}}
3085:(because given a singleton
1801:for every compact subspace
585:Since all the functions in
534:with respect to the family
207:for every compact subspace
6888:
6822:Compactly generated spaces
6778:Willard, Stephen (2004) .
6347:Ma, Dan (19 August 2010).
5500:Otherwise the topology on
4172:. For example, the space
3710:one-point compactification
3473:one-point compactification
3227:are compactly generated.
3068:Alexandrov-discrete spaces
2602:is open (resp. closed) in
2556:is open (resp. closed) in
2466:one-point compactification
2323:is open (resp. closed) in
2264:is open (resp. closed) in
1781:is open (resp. closed) in
1735:is open (resp. closed) in
1361:is open (resp. closed) in
1315:is open (resp. closed) in
6461:Lawson & Madison 1974
6401:Lawson & Madison 1974
6020:Countably generated space
5356:Denote this new space by
5142:we can define a possibly
4802:{\displaystyle u:K\to X.}
4400:{\displaystyle X\times Y}
4367:locally compact Hausdorff
4337:{\displaystyle X\times Y}
4067:locally compact Hausdorff
3868:{\displaystyle q^{-1}(U)}
3668:first uncountable ordinal
3217:locally compact Hausdorff
3062:are CG-2. This includes
2909:cartesian closed category
2688:{\displaystyle X=\{0,1\}}
2451:locally compact Hausdorff
2449:is a quotient space of a
2422:is a quotient space of a
2391:{\displaystyle f:K\to X.}
2363:and every continuous map
2316:{\displaystyle f^{-1}(A)}
1966:is a quotient space of a
1539:) is to find a subfamily
1462:compactly generated space
777:{\displaystyle f^{-1}(U)}
399:compactly generated space
293:cartesian closed category
83:compactly generated space
59:compactly generated space
6606:Lamartin, W. F. (1977),
6299:Steen & Seebach 1995
6029:Finitely generated space
5920:{\displaystyle k(Y^{X})}
5651:the full subcategory of
5584:the full subcategory of
5349:{\displaystyle \alpha .}
4479:{\displaystyle f:X\to Y}
4109:obtained by identifying
3763:{\displaystyle q:Y\to X}
3314:on an uncountable space.
2944:de Vries duality theorem
2892:, after the German word
2205:{\displaystyle f:K\to X}
2006:{\displaystyle f:K\to X}
1890:from all compact spaces
1883:{\displaystyle f:K\to X}
829:{\displaystyle f:K\to X}
6266:Bankston, Paul (1979).
6180:Hatcher, Allen (2001).
5523:is strictly finer than
4929:For topological spaces
4450:Continuity of functions
4411:. But its k-ification
4358:space is CG-1. (Here,
3538:is an open subspace of
2896:. They were studied by
2595:{\displaystyle A\cap K}
1774:{\displaystyle A\cap K}
1354:{\displaystyle A\cap K}
180:{\displaystyle A\cap K}
6727:Seebach, J. Arthur Jr.
6710:Rezk, Charles (2018).
6681:Upper Saddle River, NJ
6567:Topology and Groupoids
6415:, 5.9.1 (Corollary 2).
6301:, Example 114, p. 136.
6285:10.1215/ijm/1256048236
6241:10.2140/pjm.1980.88.35
5992:
5972:
5948:
5921:
5885:
5842:
5784:
5761:
5741:
5707:
5679:
5645:
5607:
5578:
5537:
5517:
5494:
5462:
5442:
5419:
5399:
5376:
5350:
5327:
5300:
5264:
5250:by declaring a subset
5244:
5224:
5201:
5160:
5136:
5104:
5103:{\displaystyle C(X,Y)}
5065:
5041:
5021:
5001:
5000:{\displaystyle C(X,Y)}
4966:
4943:
4916:
4887:
4845:
4831:is CG-3, the function
4825:
4803:
4768:
4748:
4707:
4693:is CG-2, the function
4687:
4665:
4636:
4591:
4577:is CG-1, the function
4571:
4548:
4528:
4510:, one can express the
4500:
4480:
4440:
4401:
4338:
4308:
4250:from the real line is
4240:
4135:
4103:
4057:
4030:
4014:weakly locally compact
3999:
3972:
3926:
3869:
3830:
3810:
3784:
3764:
3732:
3691:
3660:
3633:
3552:
3532:
3512:
3492:
3465:
3445:
3423:
3395:
3366:
3296:
3276:
3252:
3209:weakly locally compact
3183:
3163:
3140:
3114:
3077:Every CG-3 space is a
3064:first countable spaces
2946:. A definition of the
2860:
2859:{\displaystyle (0,1].}
2825:
2785:
2759:
2733:
2689:
2645:
2616:
2596:
2570:
2550:
2508:
2443:
2416:
2392:
2357:
2337:
2317:
2278:
2258:
2229:
2206:
2170:
2138:
2108:
2085:
2056:
2027:
2007:
1968:weakly locally compact
1960:
1929:
1907:
1884:
1848:
1824:
1795:
1775:
1749:
1729:
1696:
1664:
1614:
1587:
1567:
1533:
1509:
1478:
1454:
1431:
1408:
1375:
1355:
1329:
1309:
1275:
1255:
1232:
1208:
1185:
1159:
1127:
1104:
1081:
1061:
1037:
1005:
983:
957:
928:
908:
885:
857:
830:
798:
778:
739:
713:
682:
658:
638:
637:{\displaystyle (X,T),}
603:
579:
552:
528:
508:
473:
446:
423:
389:
362:
338:
270:
250:
230:
201:
181:
155:
131:
112:
51:
6388:, Proposition 3.4(3).
6015:Compact-open topology
6000:compact-open topology
5993:
5973:
5949:
5947:{\displaystyle Y^{X}}
5922:
5886:
5843:
5785:
5762:
5742:
5708:
5680:
5646:
5608:
5579:
5546:This construction is
5538:
5518:
5495:
5493:{\displaystyle kX=X.}
5463:
5443:
5420:
5400:
5377:
5351:
5328:
5301:
5265:
5245:
5225:
5202:
5174:of the topology. Let
5161:
5137:
5114:equivalence classes.
5105:
5066:
5049:compact-open topology
5042:
5022:
5002:
4967:
4944:
4917:
4888:
4846:
4826:
4804:
4769:
4749:
4708:
4688:
4666:
4637:
4592:
4572:
4549:
4529:
4501:
4481:
4441:
4402:
4339:
4309:
4241:
4136:
4134:{\displaystyle (0,1]}
4104:
4058:
4031:
4000:
3998:{\displaystyle X_{i}}
3973:
3927:
3870:
3831:
3811:
3785:
3765:
3733:
3692:
3661:
3634:
3553:
3533:
3513:
3493:
3466:
3446:
3424:
3396:
3372:(each with the usual
3367:
3297:
3277:
3253:
3225:topological manifolds
3184:
3164:
3141:
3115:
2955:identification spaces
2907:. This fails to be a
2861:
2826:
2786:
2784:{\displaystyle \{1\}}
2760:
2758:{\displaystyle \{0\}}
2734:
2690:
2646:
2617:
2597:
2571:
2551:
2509:
2479:property to form the
2444:
2417:
2393:
2358:
2338:
2318:
2279:
2259:
2230:
2207:
2171:
2139:
2109:
2086:
2057:
2028:
2013:from a compact space
2008:
1961:
1930:
1908:
1885:
1849:
1825:
1796:
1776:
1750:
1730:
1697:
1665:
1615:
1588:
1568:
1534:
1510:
1479:
1455:
1432:
1409:
1376:
1356:
1330:
1310:
1276:
1256:
1233:
1209:
1186:
1160:
1158:{\displaystyle kX=X.}
1128:
1105:
1082:
1062:
1038:
1006:
984:
958:
929:
909:
886:
858:
831:
799:
779:
740:
714:
683:
659:
639:
609:were continuous into
604:
580:
553:
529:
509:
474:
447:
424:
390:
363:
339:
337:{\displaystyle (X,T)}
271:
251:
231:
202:
182:
156:
132:
113:
52:
6427:, Proposition 5.9.1.
5982:
5962:
5931:
5895:
5858:
5801:
5771:
5751:
5717:
5689:
5655:
5618:
5589:
5554:
5527:
5504:
5472:
5452:
5429:
5409:
5386:
5360:
5337:
5310:
5277:
5254:
5234:
5211:
5178:
5150:
5126:
5079:
5055:
5031:
5011:
4976:
4953:
4933:
4897:
4855:
4835:
4815:
4778:
4758:
4720:
4697:
4677:
4646:
4604:
4581:
4561:
4538:
4518:
4490:
4458:
4415:
4385:
4322:
4258:
4176:
4144:More generally, any
4113:
4081:
4044:
4020:
3982:
3936:
3897:
3840:
3820:
3794:
3790:and for an open set
3774:
3742:
3719:
3674:
3643:
3585:
3542:
3522:
3502:
3479:
3455:
3433:
3413:
3383:
3354:
3312:cocountable topology
3286:
3266:
3258:is anticompact and T
3242:
3173:
3150:
3124:
3089:
2980:As explained in the
2835:
2803:
2769:
2743:
2699:
2661:
2626:
2606:
2580:
2560:
2534:
2498:
2494:A topological space
2457:As explained in the
2433:
2406:
2367:
2347:
2327:
2288:
2268:
2242:
2216:
2184:
2160:
2156:(1) The topology on
2128:
2124:A topological space
2095:
2084:{\displaystyle f(K)}
2066:
2055:{\displaystyle f(K)}
2037:
2033:has a compact image
2017:
1985:
1974:As explained in the
1950:
1919:
1894:
1862:
1838:
1834:(2) The topology on
1805:
1785:
1759:
1739:
1713:
1686:
1682:(1) The topology on
1654:
1650:A topological space
1597:
1577:
1543:
1519:
1499:
1468:
1444:
1418:
1385:
1365:
1339:
1319:
1293:
1265:
1242:
1218:
1195:
1171:
1137:
1114:
1091:
1071:
1047:
1023:
995:
967:
938:
918:
895:
875:
840:
808:
788:
749:
723:
700:
672:
648:
613:
589:
566:
538:
518:
489:
456:
433:
409:
376:
352:
316:
260:
240:
211:
191:
165:
145:
121:
93:
76:weak Hausdorff space
41:
6679:(Second ed.).
6535:, Proposition 1.11.
6499:, Proposition 1.11.
5375:{\displaystyle kX.}
5047:topologized by the
4774:and continuous map
4409:categorical product
3816:the restriction of
3046:without ambiguity.
2917:identification maps
2522:if its topology is
2516:compactly-generated
2176:coincides with the
2146:compactly-generated
1854:coincides with the
1672:compactly-generated
1283:compactly generated
1013:compactly generated
982:{\displaystyle kX.}
963:is usually denoted
644:the k-ification of
479:as detailed below.
85:is a space that is
6790:Dover Publications
6723:Steen, Lynn Arthur
6685:Prentice Hall, Inc
6618:Mac Lane, Saunders
6585:Engelking, Ryszard
6487:, Proposition 7.5.
6475:, Proposition 2.6.
6463:, Proposition 1.2.
6439:, Proposition 1.7.
6376:, Proposition 2.2.
6364:, Proposition 1.8.
6256:, Proposition 1.6.
6183:Algebraic Topology
6098:, Definition 43.8.
6083:10.1007/BF02194829
5988:
5968:
5944:
5917:
5881:
5852:exponential object
5838:
5783:{\displaystyle kX}
5780:
5757:
5737:
5703:
5675:
5641:
5603:
5574:
5533:
5516:{\displaystyle kX}
5513:
5490:
5458:
5441:{\displaystyle kX}
5438:
5415:
5398:{\displaystyle kX}
5395:
5372:
5346:
5323:
5296:
5260:
5240:
5223:{\displaystyle X.}
5220:
5197:
5156:
5132:
5110:are precisely the
5100:
5061:
5037:
5017:
4997:
4965:{\displaystyle Y,}
4962:
4939:
4912:
4883:
4841:
4821:
4799:
4764:
4744:
4703:
4683:
4661:
4632:
4587:
4567:
4544:
4524:
4496:
4476:
4436:
4397:
4373:When working in a
4334:
4304:
4236:
4131:
4099:
4056:{\displaystyle X.}
4053:
4026:
3995:
3968:
3922:
3865:
3826:
3806:
3780:
3760:
3731:{\displaystyle X,}
3728:
3687:
3656:
3629:
3558:that is not CG-2.
3548:
3528:
3508:
3491:{\displaystyle Y.}
3488:
3461:
3441:
3419:
3391:
3374:Euclidean topology
3362:
3292:
3272:
3248:
3179:
3162:{\displaystyle K;}
3159:
3136:
3110:
2962:algebraic topology
2948:exponential object
2856:
2821:
2781:
2755:
2729:
2685:
2641:
2612:
2592:
2566:
2546:
2504:
2439:
2412:
2388:
2353:
2333:
2313:
2274:
2254:
2228:{\displaystyle K.}
2225:
2202:
2166:
2134:
2107:{\displaystyle X.}
2104:
2081:
2052:
2023:
2003:
1956:
1943:of compact spaces.
1925:
1906:{\displaystyle K.}
1903:
1880:
1844:
1820:
1791:
1771:
1745:
1725:
1692:
1660:
1610:
1583:
1563:
1529:
1505:
1474:
1450:
1430:{\displaystyle X,}
1427:
1404:
1371:
1351:
1325:
1305:
1271:
1254:{\displaystyle X.}
1251:
1228:
1207:{\displaystyle X,}
1204:
1181:
1155:
1126:{\displaystyle X;}
1123:
1103:{\displaystyle X,}
1100:
1077:
1057:
1033:
1001:
979:
953:
924:
907:{\displaystyle X,}
904:
881:
853:
826:
794:
774:
735:
719:they are the sets
712:{\displaystyle X;}
709:
678:
654:
634:
599:
578:{\displaystyle T.}
575:
548:
524:
504:
469:
445:{\displaystyle X.}
442:
419:
388:{\displaystyle X.}
385:
358:
334:
297:algebraic topology
266:
246:
226:
197:
177:
151:
127:
108:
47:
6799:978-0-486-43479-7
6750:978-0-486-68735-3
6694:978-0-13-181629-9
6673:Munkres, James R.
6547:, Problem 43J(1).
6451:, Example 3.3.29.
6337:, Example 1.6.19.
6313:, Problem 43H(2).
6059:, Definition 1.1.
5991:{\displaystyle Y}
5971:{\displaystyle X}
5796:inclusion functor
5760:{\displaystyle X}
5536:{\displaystyle X}
5461:{\displaystyle X}
5418:{\displaystyle X}
5263:{\displaystyle A}
5243:{\displaystyle X}
5159:{\displaystyle X}
5135:{\displaystyle X}
5064:{\displaystyle X}
5040:{\displaystyle Y}
5020:{\displaystyle X}
4942:{\displaystyle X}
4844:{\displaystyle f}
4824:{\displaystyle X}
4767:{\displaystyle K}
4706:{\displaystyle f}
4686:{\displaystyle X}
4590:{\displaystyle f}
4570:{\displaystyle X}
4547:{\displaystyle f}
4527:{\displaystyle f}
4499:{\displaystyle X}
4316:quotient topology
4248:subspace topology
4029:{\displaystyle X}
3829:{\displaystyle q}
3783:{\displaystyle Y}
3712:, which is CG-1.
3551:{\displaystyle X}
3531:{\displaystyle Y}
3511:{\displaystyle X}
3464:{\displaystyle X}
3422:{\displaystyle Y}
3403:discrete topology
3295:{\displaystyle X}
3275:{\displaystyle X}
3251:{\displaystyle X}
3195:discrete topology
3182:{\displaystyle X}
3060:Sequential spaces
3040:
3039:
2913:cartesian product
2793:discrete topology
2615:{\displaystyle K}
2569:{\displaystyle X}
2507:{\displaystyle X}
2442:{\displaystyle X}
2415:{\displaystyle X}
2356:{\displaystyle K}
2336:{\displaystyle K}
2277:{\displaystyle X}
2169:{\displaystyle X}
2137:{\displaystyle X}
2026:{\displaystyle K}
1959:{\displaystyle X}
1928:{\displaystyle X}
1847:{\displaystyle X}
1794:{\displaystyle K}
1748:{\displaystyle X}
1695:{\displaystyle X}
1663:{\displaystyle X}
1586:{\displaystyle X}
1508:{\displaystyle X}
1477:{\displaystyle X}
1453:{\displaystyle X}
1374:{\displaystyle K}
1328:{\displaystyle X}
1274:{\displaystyle X}
1080:{\displaystyle X}
1004:{\displaystyle X}
927:{\displaystyle X}
884:{\displaystyle X}
797:{\displaystyle K}
681:{\displaystyle T}
657:{\displaystyle T}
527:{\displaystyle X}
361:{\displaystyle T}
346:topological space
269:{\displaystyle X}
249:{\displaystyle X}
200:{\displaystyle K}
154:{\displaystyle X}
130:{\displaystyle A}
50:{\displaystyle X}
36:topological space
16:(Redirected from
6879:
6867:General topology
6811:
6781:General Topology
6774:
6772:
6762:
6718:
6716:
6706:
6668:
6656:
6641:
6613:
6602:
6589:General Topology
6580:
6548:
6542:
6536:
6530:
6524:
6523:, Theorem 43.10.
6518:
6512:
6506:
6500:
6494:
6488:
6482:
6476:
6470:
6464:
6458:
6452:
6446:
6440:
6434:
6428:
6422:
6416:
6410:
6404:
6398:
6389:
6383:
6377:
6371:
6365:
6359:
6353:
6352:
6344:
6338:
6332:
6326:
6320:
6314:
6308:
6302:
6296:
6290:
6289:
6287:
6263:
6257:
6251:
6245:
6244:
6226:
6217:
6211:
6205:
6196:
6193:See the Appendix
6190:
6188:
6177:
6171:
6165:
6156:
6144:
6138:
6132:
6126:
6120:
6111:
6105:
6099:
6093:
6087:
6086:
6066:
6060:
6054:
6034:
6025:
5997:
5995:
5994:
5989:
5977:
5975:
5974:
5969:
5954:is the space of
5953:
5951:
5950:
5945:
5943:
5942:
5926:
5924:
5923:
5918:
5913:
5912:
5890:
5888:
5887:
5882:
5880:
5847:
5845:
5844:
5839:
5834:
5820:
5789:
5787:
5786:
5781:
5766:
5764:
5763:
5758:
5746:
5744:
5743:
5738:
5736:
5712:
5710:
5709:
5704:
5702:
5684:
5682:
5681:
5676:
5674:
5650:
5648:
5647:
5642:
5640:
5612:
5610:
5609:
5604:
5602:
5583:
5581:
5580:
5575:
5573:
5542:
5540:
5539:
5534:
5522:
5520:
5519:
5514:
5499:
5497:
5496:
5491:
5467:
5465:
5464:
5459:
5447:
5445:
5444:
5439:
5424:
5422:
5421:
5416:
5404:
5402:
5401:
5396:
5381:
5379:
5378:
5373:
5355:
5353:
5352:
5347:
5332:
5330:
5329:
5324:
5322:
5321:
5305:
5303:
5302:
5297:
5295:
5294:
5269:
5267:
5266:
5261:
5249:
5247:
5246:
5241:
5229:
5227:
5226:
5221:
5206:
5204:
5203:
5198:
5193:
5192:
5172:
5171:
5165:
5163:
5162:
5157:
5141:
5139:
5138:
5133:
5109:
5107:
5106:
5101:
5070:
5068:
5067:
5062:
5046:
5044:
5043:
5038:
5026:
5024:
5023:
5018:
5006:
5004:
5003:
4998:
4971:
4969:
4968:
4963:
4948:
4946:
4945:
4940:
4921:
4919:
4918:
4913:
4892:
4890:
4889:
4884:
4870:
4869:
4850:
4848:
4847:
4842:
4830:
4828:
4827:
4822:
4808:
4806:
4805:
4800:
4773:
4771:
4770:
4765:
4753:
4751:
4750:
4745:
4712:
4710:
4709:
4704:
4692:
4690:
4689:
4684:
4670:
4668:
4667:
4662:
4641:
4639:
4638:
4633:
4619:
4618:
4596:
4594:
4593:
4588:
4576:
4574:
4573:
4568:
4553:
4551:
4550:
4545:
4533:
4531:
4530:
4525:
4505:
4503:
4502:
4497:
4485:
4483:
4482:
4477:
4445:
4443:
4442:
4437:
4406:
4404:
4403:
4398:
4379:product topology
4343:
4341:
4340:
4335:
4313:
4311:
4310:
4305:
4276:
4271:
4245:
4243:
4242:
4237:
4223:
4209:
4189:
4140:
4138:
4137:
4132:
4108:
4106:
4105:
4102:{\displaystyle }
4100:
4075:Sierpiński space
4062:
4060:
4059:
4054:
4035:
4033:
4032:
4027:
4004:
4002:
4001:
3996:
3994:
3993:
3977:
3975:
3974:
3969:
3967:
3966:
3951:
3950:
3931:
3929:
3928:
3923:
3921:
3920:
3911:
3910:
3905:
3874:
3872:
3871:
3866:
3855:
3854:
3835:
3833:
3832:
3827:
3815:
3813:
3812:
3807:
3789:
3787:
3786:
3781:
3769:
3767:
3766:
3761:
3737:
3735:
3734:
3729:
3715:In a CG-2 space
3703:Arens-Fort space
3699:Fortissimo space
3696:
3694:
3693:
3688:
3686:
3685:
3665:
3663:
3662:
3657:
3655:
3654:
3638:
3636:
3635:
3630:
3625:
3624:
3597:
3596:
3557:
3555:
3554:
3549:
3537:
3535:
3534:
3529:
3517:
3515:
3514:
3509:
3497:
3495:
3494:
3489:
3470:
3468:
3467:
3462:
3450:
3448:
3447:
3442:
3440:
3428:
3426:
3425:
3420:
3400:
3398:
3397:
3392:
3390:
3371:
3369:
3368:
3363:
3361:
3326:Arens-Fort space
3319:Fortissimo space
3301:
3299:
3298:
3293:
3281:
3279:
3278:
3273:
3257:
3255:
3254:
3249:
3199:Sierpiński space
3193:spaces have the
3188:
3186:
3185:
3180:
3168:
3166:
3165:
3160:
3145:
3143:
3142:
3137:
3119:
3117:
3116:
3111:
3006:Meaning summary
3000:
2940:full subcategory
2865:
2863:
2862:
2857:
2830:
2828:
2827:
2824:{\displaystyle }
2822:
2790:
2788:
2787:
2782:
2764:
2762:
2761:
2756:
2738:
2736:
2735:
2730:
2694:
2692:
2691:
2686:
2656:Sierpiński space
2650:
2648:
2647:
2642:
2621:
2619:
2618:
2613:
2601:
2599:
2598:
2593:
2575:
2573:
2572:
2567:
2555:
2553:
2552:
2547:
2513:
2511:
2510:
2505:
2470:Arens-Fort space
2448:
2446:
2445:
2440:
2421:
2419:
2418:
2413:
2397:
2395:
2394:
2389:
2362:
2360:
2359:
2354:
2342:
2340:
2339:
2334:
2322:
2320:
2319:
2314:
2303:
2302:
2283:
2281:
2280:
2275:
2263:
2261:
2260:
2255:
2234:
2232:
2231:
2226:
2211:
2209:
2208:
2203:
2175:
2173:
2172:
2167:
2143:
2141:
2140:
2135:
2113:
2111:
2110:
2105:
2090:
2088:
2087:
2082:
2061:
2059:
2058:
2053:
2032:
2030:
2029:
2024:
2012:
2010:
2009:
2004:
1965:
1963:
1962:
1957:
1934:
1932:
1931:
1926:
1912:
1910:
1909:
1904:
1889:
1887:
1886:
1881:
1853:
1851:
1850:
1845:
1829:
1827:
1826:
1821:
1800:
1798:
1797:
1792:
1780:
1778:
1777:
1772:
1754:
1752:
1751:
1746:
1734:
1732:
1731:
1726:
1701:
1699:
1698:
1693:
1669:
1667:
1666:
1661:
1633:
1632:
1619:
1617:
1616:
1611:
1606:
1605:
1592:
1590:
1589:
1584:
1572:
1570:
1569:
1564:
1562:
1561:
1552:
1551:
1538:
1536:
1535:
1530:
1528:
1527:
1514:
1512:
1511:
1506:
1483:
1481:
1480:
1475:
1459:
1457:
1456:
1451:
1436:
1434:
1433:
1428:
1413:
1411:
1410:
1405:
1400:
1399:
1380:
1378:
1377:
1372:
1360:
1358:
1357:
1352:
1334:
1332:
1331:
1326:
1314:
1312:
1311:
1306:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1238:of subspaces of
1237:
1235:
1234:
1229:
1227:
1226:
1213:
1211:
1210:
1205:
1190:
1188:
1187:
1182:
1180:
1179:
1164:
1162:
1161:
1156:
1133:or in short, if
1132:
1130:
1129:
1124:
1109:
1107:
1106:
1101:
1086:
1084:
1083:
1078:
1066:
1064:
1063:
1058:
1056:
1055:
1042:
1040:
1039:
1034:
1032:
1031:
1010:
1008:
1007:
1002:
988:
986:
985:
980:
962:
960:
959:
954:
952:
951:
950:
933:
931:
930:
925:
913:
911:
910:
905:
890:
888:
887:
882:
869:
868:
862:
860:
859:
854:
849:
848:
835:
833:
832:
827:
803:
801:
800:
795:
783:
781:
780:
775:
764:
763:
744:
742:
741:
736:
718:
716:
715:
710:
694:
693:
687:
685:
684:
679:
663:
661:
660:
655:
643:
641:
640:
635:
608:
606:
605:
600:
598:
597:
584:
582:
581:
576:
557:
555:
554:
549:
547:
546:
533:
531:
530:
525:
513:
511:
510:
505:
503:
502:
501:
478:
476:
475:
470:
465:
464:
451:
449:
448:
443:
428:
426:
425:
420:
418:
417:
394:
392:
391:
386:
367:
365:
364:
359:
343:
341:
340:
335:
282:Hausdorff spaces
275:
273:
272:
267:
255:
253:
252:
247:
235:
233:
232:
227:
206:
204:
203:
198:
186:
184:
183:
178:
160:
158:
157:
152:
136:
134:
133:
128:
117:
115:
114:
109:
56:
54:
53:
48:
21:
6887:
6886:
6882:
6881:
6880:
6878:
6877:
6876:
6872:Homotopy theory
6857:
6856:
6818:
6816:Further reading
6800:
6777:
6770:
6765:
6751:
6741:Springer-Verlag
6721:
6714:
6709:
6695:
6671:
6665:
6654:
6644:
6638:
6616:
6605:
6599:
6583:
6578:
6560:
6557:
6552:
6551:
6543:
6539:
6533:Strickland 2009
6531:
6527:
6519:
6515:
6507:
6503:
6495:
6491:
6483:
6479:
6473:Strickland 2009
6471:
6467:
6459:
6455:
6447:
6443:
6435:
6431:
6423:
6419:
6411:
6407:
6399:
6392:
6384:
6380:
6374:Strickland 2009
6372:
6368:
6360:
6356:
6346:
6345:
6341:
6333:
6329:
6321:
6317:
6309:
6305:
6297:
6293:
6265:
6264:
6260:
6254:Strickland 2009
6252:
6248:
6224:
6219:
6218:
6214:
6206:
6199:
6186:
6179:
6178:
6174:
6170:, Lemma 1.4(c).
6168:Strickland 2009
6166:
6159:
6145:
6141:
6135:Strickland 2009
6133:
6129:
6121:
6114:
6106:
6102:
6094:
6090:
6071:Semigroup Forum
6068:
6067:
6063:
6057:Strickland 2009
6055:
6051:
6046:
6032:
6023:
6011:
5980:
5979:
5960:
5959:
5956:continuous maps
5934:
5929:
5928:
5904:
5893:
5892:
5856:
5855:
5799:
5798:
5769:
5768:
5749:
5748:
5715:
5714:
5687:
5686:
5653:
5652:
5616:
5615:
5587:
5586:
5552:
5551:
5525:
5524:
5502:
5501:
5470:
5469:
5450:
5449:
5427:
5426:
5407:
5406:
5384:
5383:
5358:
5357:
5335:
5334:
5333:for each index
5313:
5308:
5307:
5286:
5275:
5274:
5252:
5251:
5232:
5231:
5209:
5208:
5184:
5176:
5175:
5169:
5168:
5148:
5147:
5124:
5123:
5120:
5077:
5076:
5073:path components
5053:
5052:
5029:
5028:
5009:
5008:
4974:
4973:
4951:
4950:
4931:
4930:
4927:
4895:
4894:
4861:
4853:
4852:
4833:
4832:
4813:
4812:
4776:
4775:
4756:
4755:
4718:
4717:
4695:
4694:
4675:
4674:
4644:
4643:
4610:
4602:
4601:
4579:
4578:
4559:
4558:
4536:
4535:
4516:
4515:
4488:
4487:
4456:
4455:
4452:
4413:
4412:
4383:
4382:
4360:locally compact
4356:locally compact
4320:
4319:
4256:
4255:
4252:first countable
4174:
4173:
4162:
4111:
4110:
4079:
4078:
4042:
4041:
4018:
4017:
3985:
3980:
3979:
3952:
3942:
3934:
3933:
3912:
3900:
3895:
3894:
3888:
3843:
3838:
3837:
3818:
3817:
3792:
3791:
3772:
3771:
3740:
3739:
3717:
3716:
3677:
3672:
3671:
3646:
3641:
3640:
3616:
3588:
3583:
3582:
3576:
3564:
3540:
3539:
3520:
3519:
3500:
3499:
3477:
3476:
3453:
3452:
3431:
3430:
3411:
3410:
3401:(each with the
3381:
3380:
3352:
3351:
3305:
3284:
3283:
3264:
3263:
3261:
3240:
3239:
3192:
3171:
3170:
3148:
3147:
3122:
3121:
3087:
3086:
3082:
2978:
2886:
2833:
2832:
2801:
2800:
2767:
2766:
2741:
2740:
2697:
2696:
2659:
2658:
2624:
2623:
2604:
2603:
2578:
2577:
2558:
2557:
2532:
2531:
2496:
2495:
2489:
2431:
2430:
2424:topological sum
2404:
2403:
2365:
2364:
2345:
2344:
2325:
2324:
2291:
2286:
2285:
2266:
2265:
2240:
2239:
2214:
2213:
2182:
2181:
2158:
2157:
2126:
2125:
2119:
2093:
2092:
2064:
2063:
2035:
2034:
2015:
2014:
1983:
1982:
1948:
1947:
1941:topological sum
1917:
1916:
1892:
1891:
1860:
1859:
1836:
1835:
1803:
1802:
1783:
1782:
1757:
1756:
1737:
1736:
1711:
1710:
1684:
1683:
1652:
1651:
1645:
1630:
1629:
1595:
1594:
1575:
1574:
1541:
1540:
1517:
1516:
1497:
1496:
1466:
1465:
1442:
1441:
1416:
1415:
1383:
1382:
1363:
1362:
1337:
1336:
1317:
1316:
1291:
1290:
1263:
1262:
1240:
1239:
1216:
1215:
1193:
1192:
1169:
1168:
1135:
1134:
1112:
1111:
1089:
1088:
1069:
1068:
1045:
1044:
1021:
1020:
993:
992:
965:
964:
941:
936:
935:
916:
915:
893:
892:
873:
872:
866:
865:
863:Similarly, the
838:
837:
806:
805:
786:
785:
752:
747:
746:
721:
720:
698:
697:
691:
690:
670:
669:
646:
645:
611:
610:
587:
586:
564:
563:
536:
535:
516:
515:
492:
487:
486:
454:
453:
431:
430:
407:
406:
374:
373:
350:
349:
314:
313:
310:
305:
258:
257:
238:
237:
209:
208:
189:
188:
163:
162:
161:if and only if
143:
142:
119:
118:
91:
90:
72:Hausdorff space
39:
38:
28:
23:
22:
15:
12:
11:
5:
6885:
6883:
6875:
6874:
6869:
6859:
6858:
6855:
6854:
6849:
6837:
6825:
6817:
6814:
6813:
6812:
6798:
6775:
6763:
6749:
6719:
6707:
6693:
6669:
6663:
6642:
6636:
6614:
6603:
6597:
6581:
6576:
6556:
6553:
6550:
6549:
6537:
6525:
6513:
6511:, section 3.5.
6501:
6489:
6477:
6465:
6453:
6449:Engelking 1989
6441:
6429:
6417:
6405:
6390:
6378:
6366:
6354:
6339:
6335:Engelking 1989
6327:
6315:
6303:
6291:
6278:(2): 241–252.
6258:
6246:
6212:
6210:, section 5.9.
6197:
6172:
6157:
6139:
6127:
6125:, p. 182.
6112:
6110:, p. 283.
6100:
6088:
6061:
6048:
6047:
6045:
6042:
6041:
6040:
6035:
6026:
6017:
6010:
6007:
5987:
5967:
5941:
5937:
5916:
5911:
5907:
5903:
5900:
5879:
5876:
5873:
5870:
5867:
5864:
5837:
5833:
5830:
5827:
5823:
5819:
5816:
5813:
5810:
5807:
5779:
5776:
5756:
5735:
5732:
5729:
5726:
5723:
5701:
5698:
5695:
5673:
5670:
5667:
5664:
5661:
5639:
5636:
5633:
5630:
5627:
5624:
5601:
5598:
5595:
5572:
5569:
5566:
5563:
5560:
5532:
5512:
5509:
5489:
5486:
5483:
5480:
5477:
5457:
5437:
5434:
5414:
5394:
5391:
5371:
5368:
5365:
5345:
5342:
5320:
5316:
5293:
5289:
5285:
5282:
5272:if and only if
5259:
5239:
5219:
5216:
5196:
5191:
5187:
5183:
5155:
5144:finer topology
5131:
5119:
5116:
5099:
5096:
5093:
5090:
5087:
5084:
5060:
5036:
5016:
4996:
4993:
4990:
4987:
4984:
4981:
4961:
4958:
4938:
4926:
4923:
4911:
4908:
4905:
4902:
4882:
4879:
4876:
4873:
4868:
4864:
4860:
4840:
4820:
4798:
4795:
4792:
4789:
4786:
4783:
4763:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4702:
4682:
4660:
4657:
4654:
4651:
4631:
4628:
4625:
4622:
4617:
4613:
4609:
4586:
4566:
4543:
4523:
4508:final topology
4495:
4475:
4472:
4469:
4466:
4463:
4451:
4448:
4435:
4432:
4429:
4426:
4423:
4420:
4396:
4393:
4390:
4371:
4370:
4369:space is CG-2.
4363:
4352:
4333:
4330:
4327:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4275:
4270:
4266:
4263:
4235:
4232:
4229:
4226:
4222:
4218:
4215:
4212:
4208:
4204:
4201:
4198:
4195:
4192:
4188:
4184:
4181:
4161:
4158:
4146:final topology
4130:
4127:
4124:
4121:
4118:
4098:
4095:
4092:
4089:
4086:
4052:
4049:
4038:disjoint union
4025:
4010:quotient space
3992:
3988:
3965:
3962:
3959:
3955:
3949:
3945:
3941:
3919:
3915:
3909:
3904:
3892:disjoint union
3887:
3884:
3877:locally closed
3864:
3861:
3858:
3853:
3850:
3846:
3825:
3805:
3802:
3799:
3779:
3759:
3756:
3753:
3750:
3747:
3727:
3724:
3684:
3680:
3653:
3649:
3628:
3623:
3619:
3615:
3612:
3609:
3606:
3603:
3600:
3595:
3591:
3575:
3572:
3563:
3560:
3547:
3527:
3507:
3487:
3484:
3460:
3439:
3418:
3407:
3406:
3389:
3377:
3360:
3340:
3339:
3336:
3329:
3322:
3315:
3303:
3291:
3271:
3259:
3247:
3205:Compact spaces
3190:
3178:
3158:
3155:
3135:
3132:
3129:
3109:
3106:
3103:
3100:
3097:
3094:
3080:
3054:weak Hausdorff
3038:
3037:
3034:
3028:
3027:
3024:
3018:
3017:
3014:
3008:
3007:
3004:
2977:
2974:
2966:weak Hausdorff
2960:In modern-day
2885:
2882:
2874:weak Hausdorff
2855:
2852:
2849:
2846:
2843:
2840:
2820:
2817:
2814:
2811:
2808:
2780:
2777:
2774:
2754:
2751:
2748:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2695:with topology
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2652:
2651:
2640:
2637:
2634:
2631:
2611:
2591:
2588:
2585:
2565:
2545:
2542:
2539:
2503:
2488:
2485:
2477:weak Hausdorff
2459:final topology
2455:
2454:
2438:
2427:
2411:
2400:
2399:
2398:
2387:
2384:
2381:
2378:
2375:
2372:
2352:
2332:
2312:
2309:
2306:
2301:
2298:
2294:
2273:
2253:
2250:
2247:
2224:
2221:
2201:
2198:
2195:
2192:
2189:
2178:final topology
2165:
2133:
2118:
2115:
2103:
2100:
2080:
2077:
2074:
2071:
2051:
2048:
2045:
2042:
2022:
2002:
1999:
1996:
1993:
1990:
1976:final topology
1972:
1971:
1955:
1944:
1937:quotient space
1924:
1913:
1902:
1899:
1879:
1876:
1873:
1870:
1867:
1856:final topology
1843:
1832:
1831:
1830:
1819:
1816:
1813:
1810:
1790:
1770:
1767:
1764:
1744:
1724:
1721:
1718:
1691:
1659:
1644:
1641:
1609:
1604:
1582:
1560:
1555:
1550:
1526:
1504:
1490:weak Hausdorff
1473:
1449:
1426:
1423:
1403:
1398:
1393:
1390:
1370:
1350:
1347:
1344:
1324:
1304:
1301:
1298:
1270:
1250:
1247:
1225:
1203:
1200:
1178:
1154:
1151:
1148:
1145:
1142:
1122:
1119:
1099:
1096:
1076:
1054:
1030:
1000:
978:
975:
972:
949:
944:
923:
903:
900:
880:
852:
847:
825:
822:
819:
816:
813:
793:
773:
770:
767:
762:
759:
755:
734:
731:
728:
708:
705:
677:
653:
633:
630:
627:
624:
621:
618:
596:
574:
571:
558:is called the
545:
523:
500:
495:
484:final topology
468:
463:
441:
438:
416:
384:
381:
357:
333:
330:
327:
324:
321:
309:
306:
304:
301:
278:final topology
265:
245:
225:
222:
219:
216:
196:
176:
173:
170:
150:
126:
107:
104:
101:
98:
67:compact spaces
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6884:
6873:
6870:
6868:
6865:
6864:
6862:
6853:
6850:
6848:
6846:
6841:
6838:
6836:
6834:
6829:
6826:
6823:
6820:
6819:
6815:
6809:
6805:
6801:
6795:
6791:
6787:
6786:Mineola, N.Y.
6783:
6782:
6776:
6769:
6764:
6760:
6756:
6752:
6746:
6742:
6738:
6734:
6733:
6728:
6724:
6720:
6713:
6708:
6704:
6700:
6696:
6690:
6686:
6682:
6678:
6674:
6670:
6666:
6664:0-226-51183-9
6660:
6653:
6652:
6647:
6646:May, J. Peter
6643:
6639:
6637:0-387-98403-8
6633:
6629:
6625:
6624:
6619:
6615:
6611:
6610:
6604:
6600:
6598:3-88538-006-4
6594:
6590:
6586:
6582:
6579:
6577:1-4196-2722-8
6573:
6570:, Booksurge,
6569:
6568:
6563:
6562:Brown, Ronald
6559:
6558:
6554:
6546:
6541:
6538:
6534:
6529:
6526:
6522:
6517:
6514:
6510:
6505:
6502:
6498:
6497:Lamartin 1977
6493:
6490:
6486:
6481:
6478:
6474:
6469:
6466:
6462:
6457:
6454:
6450:
6445:
6442:
6438:
6437:Lamartin 1977
6433:
6430:
6426:
6421:
6418:
6414:
6409:
6406:
6402:
6397:
6395:
6391:
6387:
6382:
6379:
6375:
6370:
6367:
6363:
6362:Lamartin 1977
6358:
6355:
6350:
6343:
6340:
6336:
6331:
6328:
6324:
6323:Lamartin 1977
6319:
6316:
6312:
6307:
6304:
6300:
6295:
6292:
6286:
6281:
6277:
6273:
6269:
6262:
6259:
6255:
6250:
6247:
6242:
6238:
6234:
6230:
6223:
6216:
6213:
6209:
6204:
6202:
6198:
6194:
6185:
6184:
6176:
6173:
6169:
6164:
6162:
6158:
6155:
6153:
6148:
6143:
6140:
6136:
6131:
6128:
6124:
6119:
6117:
6113:
6109:
6104:
6101:
6097:
6092:
6089:
6084:
6080:
6076:
6072:
6065:
6062:
6058:
6053:
6050:
6043:
6039:
6036:
6030:
6027:
6021:
6018:
6016:
6013:
6012:
6008:
6006:
6003:
6001:
5985:
5965:
5957:
5939:
5935:
5909:
5905:
5898:
5853:
5848:
5835:
5797:
5793:
5792:right adjoint
5777:
5774:
5754:
5613:
5550:. We denote
5549:
5544:
5530:
5510:
5507:
5487:
5484:
5481:
5478:
5475:
5455:
5435:
5432:
5412:
5392:
5389:
5369:
5366:
5363:
5343:
5340:
5318:
5314:
5306:is closed in
5291:
5287:
5283:
5280:
5273:
5270:to be closed
5257:
5237:
5217:
5214:
5189:
5185:
5173:
5153:
5145:
5129:
5117:
5115:
5113:
5094:
5091:
5088:
5082:
5074:
5071:is CG-1, the
5058:
5050:
5034:
5014:
4991:
4988:
4985:
4979:
4959:
4956:
4936:
4925:Miscellaneous
4924:
4922:
4909:
4906:
4903:
4900:
4880:
4874:
4871:
4866:
4858:
4838:
4818:
4809:
4796:
4793:
4787:
4784:
4781:
4761:
4741:
4735:
4732:
4729:
4726:
4723:
4716:
4700:
4680:
4671:
4658:
4655:
4652:
4649:
4629:
4623:
4620:
4615:
4607:
4600:
4584:
4564:
4555:
4541:
4521:
4513:
4509:
4493:
4473:
4467:
4464:
4461:
4449:
4447:
4430:
4427:
4424:
4418:
4410:
4394:
4391:
4388:
4380:
4376:
4368:
4364:
4361:
4357:
4353:
4350:
4349:
4348:
4345:
4331:
4328:
4325:
4317:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4273:
4264:
4261:
4253:
4249:
4230:
4227:
4224:
4220:
4216:
4213:
4210:
4206:
4202:
4199:
4196:
4182:
4179:
4171:
4167:
4159:
4157:
4155:
4150:
4147:
4142:
4125:
4122:
4119:
4093:
4090:
4087:
4076:
4070:
4068:
4063:
4050:
4047:
4039:
4023:
4015:
4011:
4006:
3990:
3986:
3963:
3960:
3957:
3947:
3943:
3917:
3913:
3907:
3902:
3893:
3885:
3883:
3880:
3878:
3859:
3851:
3848:
3844:
3823:
3803:
3800:
3797:
3777:
3757:
3751:
3748:
3745:
3725:
3722:
3713:
3711:
3706:
3704:
3700:
3682:
3678:
3669:
3651:
3647:
3621:
3617:
3613:
3610:
3604:
3601:
3598:
3593:
3589:
3581:
3580:ordinal space
3573:
3571:
3569:
3561:
3559:
3545:
3525:
3505:
3485:
3482:
3474:
3458:
3416:
3404:
3378:
3375:
3349:
3345:
3344:
3343:
3337:
3334:
3330:
3327:
3323:
3320:
3316:
3313:
3309:
3308:
3307:
3289:
3269:
3245:
3237:
3232:
3230:
3226:
3222:
3221:metric spaces
3218:
3212:
3210:
3206:
3202:
3200:
3196:
3176:
3156:
3153:
3133:
3130:
3127:
3107:
3104:
3101:
3095:
3084:
3075:
3073:
3072:finite spaces
3069:
3065:
3061:
3057:
3055:
3050:
3047:
3045:
3035:
3033:
3030:
3029:
3025:
3023:
3020:
3019:
3015:
3013:
3010:
3009:
3005:
3002:
3001:
2998:
2996:
2992:
2988:
2983:
2975:
2973:
2971:
2967:
2963:
2958:
2956:
2951:
2949:
2945:
2941:
2936:
2934:
2930:
2928:
2922:
2918:
2914:
2910:
2906:
2901:
2899:
2895:
2891:
2883:
2881:
2879:
2875:
2872:However, for
2870:
2866:
2853:
2847:
2844:
2841:
2815:
2812:
2809:
2798:
2794:
2775:
2749:
2723:
2720:
2714:
2708:
2679:
2676:
2673:
2667:
2664:
2657:
2638:
2635:
2632:
2629:
2609:
2589:
2586:
2583:
2563:
2543:
2540:
2537:
2529:
2528:
2527:
2525:
2521:
2517:
2501:
2492:
2486:
2484:
2482:
2478:
2473:
2471:
2467:
2462:
2460:
2452:
2436:
2428:
2425:
2409:
2401:
2385:
2382:
2376:
2373:
2370:
2350:
2330:
2307:
2299:
2296:
2292:
2284:exactly when
2271:
2251:
2248:
2245:
2237:
2236:
2222:
2219:
2199:
2193:
2190:
2187:
2179:
2163:
2155:
2154:
2153:
2151:
2147:
2131:
2122:
2117:Definition 2
2116:
2114:
2101:
2098:
2075:
2069:
2046:
2040:
2020:
2000:
1994:
1991:
1988:
1979:
1977:
1969:
1953:
1945:
1942:
1938:
1922:
1914:
1900:
1897:
1877:
1871:
1868:
1865:
1857:
1841:
1833:
1817:
1814:
1811:
1808:
1788:
1768:
1765:
1762:
1742:
1722:
1719:
1716:
1708:
1707:
1705:
1689:
1681:
1680:
1679:
1677:
1673:
1657:
1648:
1642:
1640:
1638:
1634:
1625:
1622:
1607:
1580:
1553:
1502:
1493:
1491:
1487:
1471:
1463:
1447:
1438:
1424:
1421:
1401:
1391:
1388:
1368:
1348:
1345:
1342:
1322:
1302:
1299:
1296:
1288:
1284:
1268:
1248:
1245:
1201:
1198:
1165:
1152:
1149:
1146:
1143:
1140:
1120:
1117:
1097:
1094:
1074:
1018:
1014:
998:
989:
976:
973:
970:
942:
921:
901:
898:
878:
870:
867:k-closed sets
850:
823:
817:
814:
811:
791:
768:
760:
757:
753:
732:
729:
726:
706:
703:
695:
675:
667:
651:
631:
625:
622:
619:
572:
569:
561:
521:
493:
485:
480:
466:
439:
436:
404:
400:
395:
382:
379:
371:
355:
347:
328:
325:
322:
307:
302:
300:
298:
294:
290:
285:
283:
279:
263:
243:
223:
220:
217:
214:
194:
174:
171:
168:
148:
140:
124:
105:
102:
99:
96:
88:
84:
79:
77:
73:
68:
64:
60:
44:
37:
33:
19:
6844:
6832:
6780:
6730:
6676:
6650:
6627:
6621:
6608:
6588:
6566:
6545:Willard 2004
6540:
6528:
6521:Willard 2004
6516:
6504:
6492:
6480:
6468:
6456:
6444:
6432:
6420:
6408:
6403:, p. 3.
6381:
6369:
6357:
6342:
6330:
6325:, p. 8.
6318:
6311:Willard 2004
6306:
6294:
6275:
6271:
6261:
6249:
6235:(1): 35–53.
6232:
6228:
6215:
6192:
6182:
6175:
6151:
6142:
6130:
6108:Munkres 2000
6103:
6096:Willard 2004
6091:
6074:
6070:
6064:
6052:
6004:
5891:is given by
5849:
5545:
5167:
5121:
4928:
4810:
4672:
4556:
4453:
4372:
4359:
4346:
4254:; the space
4163:
4151:
4143:
4141:to a point.
4071:
4064:
4007:
3932:of a family
3889:
3881:
3714:
3707:
3577:
3565:
3408:
3341:
3333:Appert space
3235:
3233:
3229:CW complexes
3213:
3203:
3189:). Finite T
3076:
3058:
3051:
3048:
3043:
3041:
3003:Abbreviation
2994:
2990:
2986:
2979:
2959:
2952:
2937:
2926:
2921:CW-complexes
2911:, the usual
2902:
2893:
2889:
2887:
2871:
2867:
2797:homeomorphic
2653:
2519:
2515:
2493:
2490:
2487:Definition 3
2474:
2463:
2456:
2149:
2145:
2123:
2120:
1980:
1973:
1675:
1671:
1649:
1646:
1643:Definition 1
1628:
1626:
1623:
1494:
1461:
1439:
1282:
1166:
1016:
1012:
990:
864:
689:
559:
481:
402:
398:
396:
311:
286:
256:and declare
82:
80:
62:
58:
57:is called a
29:
5747:that takes
5170:k-ification
5118:K-ification
4715:composition
4599:restriction
3236:anticompact
2982:Definitions
784:is open in
692:k-open sets
560:k-ification
303:Definitions
187:is open in
6861:Categories
6555:References
6425:Brown 2006
6413:Brown 2006
6208:Brown 2006
6123:Brown 2006
5548:functorial
4512:continuity
4170:sequential
3562:Properties
3498:The space
3451:) and let
2884:Motivation
2514:is called
2144:is called
1670:is called
1573:such that
1381:for every
1261:The space
1011:is called
991:The space
804:for every
745:such that
6729:(1995) .
6509:Rezk 2018
6485:Rezk 2018
6386:Rezk 2018
5998:with the
5822:→
5341:α
5319:α
5292:α
5284:∩
5190:α
4904:⊆
4878:→
4791:→
4739:→
4727:∘
4653:⊆
4627:→
4471:→
4428:×
4392:×
4329:×
4314:with the
4299:…
4246:with the
4231:…
4191:∖
4154:wedge sum
3961:∈
3903:∐
3886:Quotients
3849:−
3801:⊆
3755:→
3679:ω
3648:ω
3618:ω
3590:ω
3574:Subspaces
3566:(See the
3131:⊆
3102:⊆
2706:∅
2633:⊆
2587:∩
2541:⊆
2380:→
2297:−
2249:⊆
2197:→
1998:→
1875:→
1812:⊆
1766:∩
1720:⊆
1637:Hausdorff
1554:⊆
1486:Hausdorff
1392:∈
1346:∩
1300:⊆
821:→
758:−
730:⊆
218:⊆
172:∩
100:⊆
6703:42683260
6677:Topology
6675:(2000).
6648:(1999).
6620:(1998).
6587:(1989).
6564:(2006),
6077:: 1–18.
6009:See also
5112:homotopy
4375:category
4160:Products
3568:Examples
2976:Examples
2898:Hurewicz
2890:k-spaces
2524:coherent
1704:coherent
1287:coherent
1281:is then
370:topology
348:, where
87:coherent
32:topology
6842:at the
6830:at the
6759:0507446
6149:at the
5794:to the
4166:product
3666:is the
3471:be the
3348:product
2894:kompakt
2520:k-space
2468:of the
2150:k-space
1676:k-space
1017:k-space
403:k-space
368:is the
63:k-space
6808:115240
6806:
6796:
6757:
6747:
6701:
6691:
6661:
6634:
6595:
6574:
5927:where
5051:. If
3639:where
2530:a set
2453:space.
2238:a set
1970:space.
1709:a set
6771:(PDF)
6737:Dover
6715:(PDF)
6655:(PDF)
6225:(PDF)
6187:(PDF)
6044:Notes
5958:from
3083:space
2518:or a
2148:or a
2091:into
1939:of a
1935:is a
1674:or a
1015:or a
666:finer
344:be a
6804:OCLC
6794:ISBN
6745:ISBN
6699:OCLC
6689:ISBN
6659:ISBN
6632:ISBN
6593:ISBN
6572:ISBN
5850:The
5405:and
4972:let
4949:and
4164:The
3890:The
3346:The
3331:The
3324:The
3310:The
3223:and
3207:and
3052:For
3032:CG-3
3022:CG-2
3012:CG-1
2995:CG-3
2991:CG-2
2987:CG-1
2878:CGWH
2765:and
2429:(3)
2402:(2)
1946:(4)
1915:(3)
482:The
312:Let
139:open
34:, a
6847:Lab
6835:Lab
6280:doi
6237:doi
6154:Lab
6079:doi
5978:to
5854:in
5790:is
5767:to
5713:to
5146:on
5075:in
5027:to
4811:If
4673:If
4557:If
4514:of
4381:on
3836:to
3475:of
2931:on
2929:Lab
2915:of
1702:is
1488:or
871:in
836:in
696:in
664:is
562:of
514:on
401:or
141:in
137:is
74:or
61:or
30:In
6863::
6802:.
6792:.
6788::
6784:.
6755:MR
6753:.
6743:.
6725:;
6697:.
6687:.
6683::
6393:^
6276:23
6274:.
6270:.
6233:88
6231:.
6227:.
6200:^
6160:^
6115:^
6073:.
6002:.
4152:A
4008:A
3405:).
3376:).
3321:).
3074:.
3070:,
3066:,
2993:,
2989:,
2972:.
2935:.
2483:.
1639:.
299:.
284:.
6845:n
6833:n
6810:.
6773:.
6761:.
6735:(
6717:.
6705:.
6667:.
6640:.
6628:5
6601:.
6351:.
6288:.
6282::
6243:.
6239::
6195:)
6191:(
6189:.
6152:n
6137:.
6085:.
6081::
6075:9
5986:Y
5966:X
5940:X
5936:Y
5915:)
5910:X
5906:Y
5902:(
5899:k
5878:s
5875:u
5872:a
5869:H
5866:G
5863:C
5836:.
5832:p
5829:o
5826:T
5818:p
5815:o
5812:T
5809:G
5806:C
5778:X
5775:k
5755:X
5734:p
5731:o
5728:T
5725:G
5722:C
5700:p
5697:o
5694:T
5672:p
5669:o
5666:T
5663:G
5660:C
5638:s
5635:u
5632:a
5629:H
5626:G
5623:C
5600:p
5597:o
5594:T
5571:p
5568:o
5565:T
5562:G
5559:C
5531:X
5511:X
5508:k
5488:.
5485:X
5482:=
5479:X
5476:k
5456:X
5436:X
5433:k
5413:X
5393:X
5390:k
5370:.
5367:X
5364:k
5344:.
5315:K
5288:K
5281:A
5258:A
5238:X
5218:.
5215:X
5195:}
5186:K
5182:{
5154:X
5130:X
5098:)
5095:Y
5092:,
5089:X
5086:(
5083:C
5059:X
5035:Y
5015:X
4995:)
4992:Y
4989:,
4986:X
4983:(
4980:C
4960:,
4957:Y
4937:X
4910:.
4907:X
4901:K
4881:Y
4875:K
4872::
4867:K
4863:|
4859:f
4839:f
4819:X
4797:.
4794:X
4788:K
4785::
4782:u
4762:K
4742:Y
4736:K
4733::
4730:u
4724:f
4701:f
4681:X
4659:.
4656:X
4650:K
4630:Y
4624:K
4621::
4616:K
4612:|
4608:f
4585:f
4565:X
4542:f
4522:f
4494:X
4474:Y
4468:X
4465::
4462:f
4434:)
4431:Y
4425:X
4422:(
4419:k
4395:Y
4389:X
4332:Y
4326:X
4302:}
4296:,
4293:3
4290:,
4287:2
4284:,
4281:1
4278:{
4274:/
4269:R
4265:=
4262:Y
4234:}
4228:,
4225:3
4221:/
4217:1
4214:,
4211:2
4207:/
4203:1
4200:,
4197:1
4194:{
4187:R
4183:=
4180:X
4129:]
4126:1
4123:,
4120:0
4117:(
4097:]
4094:1
4091:,
4088:0
4085:[
4051:.
4048:X
4024:X
3991:i
3987:X
3964:I
3958:i
3954:)
3948:i
3944:X
3940:(
3918:i
3914:X
3908:i
3863:)
3860:U
3857:(
3852:1
3845:q
3824:q
3804:X
3798:U
3778:Y
3758:X
3752:Y
3749::
3746:q
3726:,
3723:X
3683:1
3652:1
3627:]
3622:1
3614:,
3611:0
3608:[
3605:=
3602:1
3599:+
3594:1
3546:X
3526:Y
3506:X
3486:.
3483:Y
3459:X
3438:R
3417:Y
3388:Z
3359:R
3335:.
3328:.
3304:1
3290:X
3270:X
3260:1
3246:X
3191:1
3177:X
3157:;
3154:K
3134:X
3128:K
3108:,
3105:X
3099:}
3096:x
3093:{
3081:1
3079:T
2927:n
2854:.
2851:]
2848:1
2845:,
2842:0
2839:(
2819:]
2816:1
2813:,
2810:0
2807:[
2779:}
2776:1
2773:{
2753:}
2750:0
2747:{
2727:}
2724:X
2721:,
2718:}
2715:1
2712:{
2709:,
2703:{
2683:}
2680:1
2677:,
2674:0
2671:{
2668:=
2665:X
2639:.
2636:X
2630:K
2610:K
2590:K
2584:A
2564:X
2544:X
2538:A
2502:X
2437:X
2410:X
2386:.
2383:X
2377:K
2374::
2371:f
2351:K
2331:K
2311:)
2308:A
2305:(
2300:1
2293:f
2272:X
2252:X
2246:A
2223:.
2220:K
2200:X
2194:K
2191::
2188:f
2164:X
2132:X
2102:.
2099:X
2079:)
2076:K
2073:(
2070:f
2050:)
2047:K
2044:(
2041:f
2021:K
2001:X
1995:K
1992::
1989:f
1954:X
1923:X
1901:.
1898:K
1878:X
1872:K
1869::
1866:f
1842:X
1818:.
1815:X
1809:K
1789:K
1769:K
1763:A
1743:X
1723:X
1717:A
1690:X
1658:X
1608:.
1603:G
1581:X
1559:F
1549:G
1525:F
1503:X
1472:X
1448:X
1425:,
1422:X
1402:.
1397:C
1389:K
1369:K
1349:K
1343:A
1323:X
1303:X
1297:A
1269:X
1249:.
1246:X
1224:C
1202:,
1199:X
1177:F
1153:.
1150:X
1147:=
1144:X
1141:k
1121:;
1118:X
1098:,
1095:X
1075:X
1053:F
1029:F
999:X
977:.
974:X
971:k
948:F
943:T
922:X
902:,
899:X
879:X
851:.
846:F
824:X
818:K
815::
812:f
792:K
772:)
769:U
766:(
761:1
754:f
733:X
727:U
707:;
704:X
676:T
652:T
632:,
629:)
626:T
623:,
620:X
617:(
595:F
573:.
570:T
544:F
522:X
499:F
494:T
467:,
462:F
440:.
437:X
415:F
383:.
380:X
356:T
332:)
329:T
326:,
323:X
320:(
264:X
244:X
224:.
221:X
215:K
195:K
175:K
169:A
149:X
125:A
106:,
103:X
97:A
45:X
20:)
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