Regular Space - Relationships To Other Separation Axioms

Relationships To Other Separation Axioms

A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated by neighbourhoods. 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_2½. (However, such a space need not be completely Hausdorff.) Thus, the definition of T₃ may cite T₀, T₁, or T_2½ 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. 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, paracompactness, 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 are regular; in fact, they are usually completely regular, which is a stronger condition. Regular spaces should also be contrasted with normal spaces.