Regular Space - Definitions

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.

A T₃ space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T₂ space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T₃ if and only if it is both regular and T₀. (A T₀ or Kolmogorov space is a topological space in which any two distinct points are topologically distinguishable, i.e., for every pair of distinct points, at least one of them has an open neighborhood not containing the other.) 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 interchangeably. In this article, we will use the term "regular" freely, but we will usually say "regular Hausdorff", which is unambiguous, 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.