760:) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, regularity is not really the issue here, and we could impose a weaker condition instead to get the same result. However, definitions are usually still phrased in terms of regularity, since this condition is more well known than any weaker one.
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There are many results for topological spaces that hold for both regular and
Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those
814:(and hence not preregular) cannot be regular. Most examples of regular and nonregular spaces studied in mathematics may be found in those two articles. On the other hand, spaces that are regular but not completely regular, or preregular but not regular, are usually constructed only to provide
845:
Most interesting spaces in mathematics that are regular also satisfy some stronger condition. Thus, regular spaces are usually studied to find properties and theorems, such as the ones below, that are actually applied to completely regular spaces, typically in analysis.
639:
is a topological space where every point has an open neighbourhood that is regular. Every regular space is locally regular, but the converse is not true. A classical example of a locally regular space that is not regular is the
608:
regular space is
Hausdorff: given two distinct points, at least one of them misses the closure of the other one, so (by regularity) there exist disjoint neighborhoods separating one point from (the closure of) the other.
620:" as they are used here, or use both terms interchangeably. This article uses the term "regular" freely, but will usually say "regular Hausdorff", which is unambiguous, instead of the less precise "T
1061:. In fact, this property characterises regular spaces; if the closed neighbourhoods of each point in a topological space form a local base at that point, then the space must be regular.
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space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T
616:" are not uncommon, there is significant variation in the literature: some authors switch the definitions of "regular" and "T
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1080:. This property is actually weaker than regularity; a topological space whose regular open sets form a base is
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with empty interior with respect to the usual
Euclidean topology, one can construct a finer topology on
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results that are truly about regularity generally don't also apply to nonregular
Hausdorff spaces.
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points can be separated by neighbourhoods. Since a
Hausdorff space is the same as a preregular
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than on regularity. An example of a regular space that is not completely regular is the
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There are many situations where another condition of topological spaces (such as
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open in the usual topology. That topology will be
Hausdorff, but not regular.
815:
771:, which is a stronger condition. Regular spaces should also be contrasted with
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733:. Thus a regular space encountered in practice can usually be assumed to be T
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not containing the other.) Indeed, if a space is
Hausdorff then it is T
596:, i.e., for every pair of distinct points, at least one of them has an
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849:
There exist
Hausdorff spaces that are not regular. An example is the
710:(Hausdorffness); all are equivalent in the context of regular spaces.
442:, represented by a dot on the left of the picture, and the closed set
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1136:"general topology - Preregular and locally compact implies regular"
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819:
721:. A space is regular if and only if its Kolmogorov quotient is T
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713:
Speaking more theoretically, the conditions of regularity and T
822:. Of course, one can easily find regular spaces that are not T
612:
Although the definitions presented here for "regular" and "T
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is a topological space in which any two distinct points are
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737:, by replacing the space with its Kolmogorov quotient.
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818:to conjectures, showing the boundaries of possible
57:. Unsourced material may be challenged and removed.
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568:is a topological space that is both regular and a
462:has plenty of room to wiggle around the open disk
1049:. In fancier terms, the closed neighbourhoods of
830:, but these examples provide more insight on the
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1068:of these closed neighbourhoods, we see that the
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403:by neighborhoods. This condition is known as
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1506:
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1021:is a regular space. Then, given any point
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117:Learn how and when to remove this message
1012:
648:Relationships to other separation axioms
534:. Concisely put, it must be possible to
433:
1102:
1076:for the open sets of the regular space
767:are regular; in fact, they are usually
580:if and only if it is both regular and T
14:
1551:
729:if and only if it's both regular and T
1153:
875:of real numbers. More generally, if
826:, and thus not Hausdorff, such as an
683:. (However, such a space need not be
652:A regular space is necessarily also
422:". These conditions are examples of
55:adding citations to reliable sources
26:
763:Most topological spaces studied in
24:
1033:, there is a closed neighbourhood
25:
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725:; and, as mentioned, a space is T
668:, a regular space which is also T
1564:Properties of topological spaces
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626:History of the separation axioms
31:
895:is a fixed nonclosed subset of
799:. Every such space is regular.
624:". For more on this issue, see
42:needs additional citations for
1128:
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429:
13:
1:
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672:must be Hausdorff (and thus T
658:topologically distinguishable
594:topologically distinguishable
545:with disjoint neighborhoods.
1179:
982:{\displaystyle U\setminus C}
932:{\displaystyle \mathbb {R} }
910:{\displaystyle \mathbb {R} }
868:{\displaystyle \mathbb {R} }
687:.) Thus, the definition of T
7:
943:the collection of all sets
418:" usually means "a regular
10:
1585:
1426:Banach fixed-point theorem
1140:Mathematics Stack Exchange
1459:
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789:small inductive dimension
572:. (A Hausdorff space or T
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804:completely regular space
802:As described above, any
779:Examples and nonexamples
507:that does not belong to
478:do not touch each other.
454:, represented by larger
564:regular Hausdorff space
18:Regular Hausdorff space
1481:Mathematics portal
1381:Metrics and properties
1367:Second-countable space
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785:zero-dimensional space
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353:and related fields of
237:(completely Hausdorff)
1013:Elementary properties
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806:is regular, and any T
765:mathematical analysis
717:-ness are related by
635:locally regular space
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387:have non-overlapping
1436:Invariance of domain
1388:Euler characteristic
1362:Bundle (mathematics)
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787:with respect to the
719:Kolmogorov quotients
685:completely Hausdorff
522:and a neighbourhood
51:improve this article
1446:Tychonoff's theorem
1441:Poincaré conjecture
1195:General (point-set)
840:Tychonoff corkscrew
255:(regular Hausdorff)
1431:De Rham cohomology
1352:Polyhedral complex
1342:Simplicial complex
1025:and neighbourhood
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810:space that is not
769:completely regular
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389:open neighborhoods
308:(completely normal
290:(normal Hausdorff)
138:topological spaces
1559:Separation axioms
1546:
1545:
1335:fundamental group
1104:Munkres, James R.
1070:regular open sets
1002:{\displaystyle U}
956:{\displaystyle U}
888:{\displaystyle C}
758:local compactness
598:open neighborhood
511:, there exists a
484:topological space
424:separation axioms
383:not contained in
359:topological space
347:
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328:(perfectly normal
133:Separation axioms
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16:(Redirected from
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1110:(2nd ed.).
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828:indiscrete space
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493:if, given any
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1408:Orientability
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1017:Suppose that
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773:normal spaces
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642:bug-eyed line
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513:neighbourhood
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68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
1538:Publications
1403:Chern number
1393:Betti number
1276: /
1267:Key concepts
1215:Differential
1139:
1130:
1107:
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1081:
1077:
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1038:
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1018:
1016:
848:
844:
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762:
743:
739:
712:
706:instead of T
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604:, and each T
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411:
410:. The term "
404:
396:
392:
384:
380:
379:and a point
376:
372:
365:
364:is called a
361:
348:
242:
225:completely T
165:(Kolmogorov)
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
1501:Wikiversity
1418:Key results
1083:semiregular
1064:Taking the
853:on the set
797:clopen sets
430:Definitions
355:mathematics
272:(Tychonoff)
201:(Hausdorff)
1553:Categories
1347:CW complex
1288:Continuity
1278:Closed set
1237:cohomology
1090:References
1055:local base
1041:that is a
851:K-topology
691:may cite T
654:preregular
495:closed set
458:. The dot
456:open disks
438:The point
145:Kolmogorov
107:April 2022
77:newspapers
1526:geometric
1521:algebraic
1372:Cobordism
1308:Hausdorff
1303:connected
1220:Geometric
1210:Continuum
1200:Algebraic
1066:interiors
974:∖
812:Hausdorff
746:normality
530:that are
401:separated
368:if every
219:(Urysohn)
183:(Fréchet)
1569:Topology
1491:Wikibook
1469:Category
1357:Manifold
1325:Homotopy
1283:Interior
1274:Open set
1232:Homology
1181:Topology
1108:Topology
1106:(2000).
820:theorems
536:separate
532:disjoint
500:and any
351:topology
1516:general
1318:uniform
1298:compact
1249:Digital
1072:form a
1053:form a
405:Axiom T
399:can be
391:. Thus
339:History
265:3½
91:scholar
1511:Topics
1313:metric
1188:Fields
1118:
1043:subset
791:has a
702:, or T
584:. (A T
470:, yet
324:
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212:½
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64:
1293:Space
836:axiom
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