Regular space

Definitions

Suppose that X is a topological space.

X is a regular space iff, given any closed set F and any point x that does not belong to F, there are a neighbourhood U of x and a neighbourhood V of F that are disjoint. In fancier terms, this condition says that x and F can be separated by neighbourhoods.

X is a T₃ space if and only if it is both regular and Hausdorff.

Note that some mathematical literature uses different definitions for the terms "regular" and "T₃". The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two terms, or use both terms synonymously for only one condition. In Wikipedia, we will use the term "regular" freely, but we'll usually say "regular Hausdorff" instead of the less clear "T₃". In other literature, you should take care to find out which definitions the author is using. (The phrase "regular Hausdorff", however, is unambiguous.) For more on this issue, see History of the separation axioms[?].

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_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 iff its Kolmogorov quotient is T₃; and, as mentioned, a space is T₃ iff 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.

Examples and nonexamples

As described above, any completely regular space 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 counterexamples to conjectures, showing the boundaries of possible theorems. Of course, one can easily find regular spaces that are not T₀, and thus not Hausdorff, such as an indiscrete space[?], but these examples provide more insight on the T₀ axiom than on regularity.

Thus, regular spaces are generally not studied because interesting spaces in mathematics are regular without also satisfying some stronger condition. Instead, they are studied to find properties and theorems, such as the ones below, that are actually applied to completely regular spaces, typically in analysis.

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 interiors of these closed neighbourhoods, we see that the regular open sets 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[?].

Extension by continuity

Suppose that A is a set in a topological space X and f is a continuous function from A to a regular space Y. Suppose that, whenever a net or filter in A converges to a point in X (say x = lim_n a_n), then f(a_n) converges to a point y in Y. Then we would like to be able to extend the domain of definition of f to the closure of A, by letting f(x) = y, and we would like the extension to be continuous as well.

If Y is a regular space, then this is always possible. If Y is regular Hausdorff, then such a continuous extension will not only exist but will be unique. Note that if A is a dense set, then f will be extended to all of X. This is called extension by continuity, since the extension of f is defined (uniquely, in the Hausdorff case) by the requirement that it be continuous.