Regular space

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 neighborhoods.

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.

A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated by neighbourhoods.

In fact, a regular Hausdorff space satisfies the slightly stronger condition T2½.

Thus, the definition of T3 may cite T0, T1, or T2½ 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 quotients.

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, or local compactness) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.

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.

A zero-dimensional space with respect to the small inductive dimension has a base consisting of clopen sets.

Most examples of regular and nonregular spaces studied in mathematics may be found in those two articles.

Most interesting spaces in mathematics that are regular also satisfy some stronger condition.