Locally regular space
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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related fields of mathematics, a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping
open neighborhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a p ...
s. Thus ''p'' and ''C'' can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
". These conditions are examples of
separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...
s.


Definitions

A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is a regular space if, given any closed set ''F'' and any point ''x'' that does not belong to ''F'', there exists a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are disjoint. Concisely put, it must be possible to separate ''x'' and ''F'' with disjoint neighborhoods. A or is a topological space that is both regular and a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if and only if it is both regular and T0. (A T0 or
Kolmogorov space In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
is a topological space in which any two distinct points are
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
, i.e., for every pair of distinct points, at least one of them has an
open neighborhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a p ...
not containing the other.) Indeed, if a space is Hausdorff then it is T0, and each T0 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 "T3" are not uncommon, there is significant variation in the literature: some authors switch the definitions of "regular" and "T3" 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 "T3". For more on this issue, see
History of the separation axioms The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, th ...
. A 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, i.e., any two
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
points can be separated by neighbourhoods. Since a Hausdorff space is the same as a preregular T0 space, a regular space which is also T0 must be Hausdorff (and thus T3). In fact, a regular Hausdorff space satisfies the slightly stronger condition T. (However, such a space need not be
completely Hausdorff In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a t ...
.) Thus, the definition of T3 may cite T0, T1, or T instead of T2 (Hausdorffness); all are equivalent in the context of regular spaces. Speaking more theoretically, the conditions of regularity and T3-ness are related by
Kolmogorov quotient In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
s. A space is regular if and only if its Kolmogorov quotient is T3; and, as mentioned, a space is T3 if and only if it's both regular and T0. Thus a regular space encountered in practice can usually be assumed to be T3, 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 normality, pseudonormality,
paracompactness In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
, or local compactness) 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. Most topological spaces studied in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
are regular; in fact, they are usually
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
, which is a stronger condition. Regular spaces should also be contrasted with
normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
s.


Examples and nonexamples

A
zero-dimensional space In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical ...
with respect to the small inductive dimension has a base consisting of
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
s. Every such space is regular. As described above, any
completely regular space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
is regular, and any T0 space that is not 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 A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
s to conjectures, showing the boundaries of possible
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
s. Of course, one can easily find regular spaces that are not T0, and thus not Hausdorff, such as an
indiscrete space In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, but these examples provide more insight on the T0 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 set R with the topology generated by sets of the form ''U — C'', where ''U'' is an open set in the usual sense, and ''C'' is any countable subset of ''U''.


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 In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
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 interiors of these closed neighbourhoods, we see that the
regular open set A subset S of a topological space X is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if \operatorname(\overline) = S or, equivalently, if \partial(\overline)=\partial S, where \operatorname S, \ov ...
s form a 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''.


References

{{Topology Separation axioms Properties of topological spaces Topology