Regular space

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In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.[1] 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". These conditions are examples of separation axioms.

Contents

Definitions

A topological space X is a regular space if, given any nonempty closed set F and any point x that does not belong to F, there exists a neighbourhood 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.

If X is both regular and Hausdorff, it is called a regular Hausdorff space or a T3 space. It turns out that a space is T3 if and only if it is both regular and T0. 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 interchangably. In this article, we will use the term "regular" freely, but we will usually say "regular Hausdorff", which is unambigous, instead of the less precise "T3". For more on this issue, see History of the separation axioms.

A 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. Since a Hausdorff space is the same as a preregular T0 space, a regular space that 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.) Thus, the definition of T3 may cite T0, T1, or T instead of T2 (Hausdorffness); all are equivalent in the context of regular spaces.

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