Editing Regular space

{{Short description|Property of topological space}}
{{More citations needed|date=April 2022}}
{{Separation axioms}}

In [[topology]] and related fields of [[mathematics]], a [[topological space]] ''X'' is called a '''regular space''' if every [[closed subset]] ''C'' of ''X'' and a point ''p'' not contained in ''C'' have non-overlapping [[open neighborhood]]s.<ref>{{cite book | last=Munkres | first=James R. | authorlink=James Munkres | title=Topology | edition=2nd | publisher=[[Prentice Hall]] | year=2000 | isbn=0-13-181629-2}}</ref> Thus ''p'' and ''C'' can be [[separated sets|separated]] by neighborhoods. This condition is known as '''Axiom T<sub>3</sub>'''. The term "'''T<sub>3</sub> space'''" usually means "a regular [[Hausdorff space]]". These conditions are examples of [[separation axiom]]s.

==Definitions==
[[Image:Regular space.svg|250px|thumb|right|The point ''x'', represented by a dot on the left of the picture, and the closed set ''F'', represented by a closed disk on the right of the picture, are separated by their neighbourhoods ''U'' and ''V'', represented by larger [[open disk]]s. The dot ''x'' has plenty of room to wiggle around the open disk ''U'', and the closed disk F has plenty of room to wiggle around the open disk ''V'', yet ''U'' and ''V'' do not touch each other.]]

A [[topological space]] ''X'' is a '''regular space''' if, given any [[closed set]] ''F'' and any [[Point (geometry)|point]] ''x'' that does not belong to ''F'', there exists a [[neighbourhood (topology)|neighbourhood]] ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are [[Disjoint sets|disjoint]]. Concisely put, it must be possible to [[separated set|separate]] ''x'' and ''F'' with disjoint neighborhoods.

A '''{{visible anchor|T3 space|text=T<sub>3</sub> space}}''' or '''{{visible anchor|regular Hausdorff space}}''' is a topological space that is both regular and a [[Hausdorff space]]. (A Hausdorff space or T<sub>2</sub> space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T<sub>3</sub> if and only if it is both regular and T<sub>0</sub>. (A T<sub>0</sub> or [[Kolmogorov space]] is a topological space in which any two distinct points are [[topologically distinguishable]], i.e., for every pair of distinct points, at least one of them has an [[open neighborhood]] not containing the other.) Indeed, if a space is Hausdorff then it is T<sub>0</sub>, and each T<sub>0</sub> 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.

Although the definitions presented here for "regular" and "T<sub>3</sub>" are not uncommon, there is significant variation in the literature: some authors switch the definitions of "regular" and "T<sub>3</sub>" 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<sub>3</sub>". For more on this issue, see [[History of the separation axioms]].

A {{em|{{visible anchor|locally regular space}}}} 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 [[bug-eyed line]].

== Relationships to other separation axioms ==
A regular space is necessarily also [[preregular space|preregular]], i.e., any two [[topologically distinguishable]] points can be separated by neighbourhoods.
Since a Hausdorff space is the same as a preregular [[Kolmogorov space|T<sub>0</sub> space]], a regular space which is also T<sub>0</sub> must be Hausdorff (and thus T<sub>3</sub>).
In fact, a regular Hausdorff space satisfies the slightly stronger condition [[Urysohn and completely Hausdorff spaces|T<sub>2½</sub>]].
(However, such a space need not be [[Completely Hausdorff space|completely Hausdorff]].)
Thus, the definition of T<sub>3</sub> may cite T<sub>0</sub>, [[T1 space|T<sub>1</sub>]], or T<sub>2½</sub> instead of T<sub>2</sub> (Hausdorffness); all are equivalent in the context of regular spaces.

Speaking more theoretically, the conditions of regularity and T<sub>3</sub>-ness are related by [[Kolmogorov quotient]]s.
A space is regular if and only if its Kolmogorov quotient is T<sub>3</sub>; and, as mentioned, a space is T<sub>3</sub> if and only if it's both regular and T<sub>0</sub>.
Thus a regular space encountered in practice can usually be assumed to be T<sub>3</sub>, by replacing the space with its Kolmogorov quotient.

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 results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.

There are many situations where another condition of topological spaces (such as [[normal space|normality]], [[Pseudonormal space|pseudonormality]], [[paracompactness]], or [[local compactness]]) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.<ref>{{cite web |title=general topology - Preregular and locally compact implies regular |url=https://math.stackexchange.com/q/1272957 |website=Mathematics Stack Exchange}}</ref>
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.

Most topological spaces studied in [[mathematical analysis]] are regular; in fact, they are usually [[completely regular space|completely regular]], which is a stronger condition.
Regular spaces should also be contrasted with [[normal space]]s.

== Examples and nonexamples ==
A [[zero-dimensional space]] with respect to the [[small inductive dimension]] has a [[Base (topology)|base]] consisting of [[clopen set]]s.
Every such space is regular.

As described above, any [[completely regular space]] is regular, and any T<sub>0</sub> space that is not [[Hausdorff space|Hausdorff]] (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 [[counterexample]]s to conjectures, showing the boundaries of possible [[theorem]]s.
Of course, one can easily find regular spaces that are not T<sub>0</sub>, and thus not Hausdorff, such as an [[indiscrete space]], but these examples provide more insight on the [[Kolmogorov axiom|T<sub>0</sub> axiom]] than on regularity.  An example of a regular space that is not completely regular is the [[Tychonoff corkscrew]].

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.

There exist Hausdorff spaces that are not regular.  An example is the [[K-topology]] on the set <math>\R</math> of real numbers.  More generally, if <math>C</math> is a fixed nonclosed subset of <math>\R</math> with empty interior with respect to the usual Euclidean topology, one can construct a finer topology on <math>\R</math> by taking as a [[base (topology)|base]] the collection of all sets <math>U</math> and <math>U\setminus C</math> for <math>U</math> open in the usual topology.  That topology will be Hausdorff, but not regular.

== Elementary properties ==
Suppose that ''X'' is a regular space.
Then, given any point ''x'' and neighbourhood ''G'' of ''x'', there is a closed neighbourhood ''E'' of ''x'' that is a [[subset]] of ''G''.
In fancier terms, the closed neighbourhoods of ''x'' form a [[local base]] at ''x''.
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.

Taking the [[interior (topology)|interior]]s of these closed neighbourhoods, we see that the [[regular open set]]s form a [[base (topology)|base]] for the open sets of the regular space ''X''.
This property is actually weaker than regularity; a topological space whose regular open sets form a base is ''[[semiregular space|semiregular]]''.

==References==
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{{Topology}}

[[Category:Separation axioms]]
[[Category:Properties of topological spaces]]
[[Category:Topology]]