Regular space

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In topology and related fields of mathematics, regular spaces and T₃ spaces are particularly special kinds of topological spaces. Both conditions are examples of separation axioms.

[edit] Definitions

The point x, represented by a dot to the left of the picture, and the closed set F, represented by a closed disk to the right of the picture, are separated by their neighbourhoods U and V, represented by larger open disks. The dot x has plenty of room to wiggle around the open disk U, and the closed disk F has plenty of room to wiggle around the open disk V, yet U and V do not touch each other.

Suppose that X is a topological space.

X is a regular space if and only if, given any 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. In fancier terms, this condition says that x and F can be separated by neighbourhoods.

X is a T₃ space or regular Hausdorff space if and only if it is both regular and Hausdorff. Also, a T₀ regular space is Hausdorff (given two points, at least one of them is not in the closure of the other one, therefore in any case they have disjoint neighborhoods). Thus it's also true that X is T₃ space if and only if it is both regular and T₀.

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 this encyclopedia, we will use the term "regular" freely, but we will usually say "regular Hausdorff" instead of the less clear "T₃". In other literature, one 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.

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.

[edit] 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 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.

[edit] Examples and nonexamples

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

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. 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 exists 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.

[edit] 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.