Editing Regular space (section)

==Definitions==
[[Image:Regular space.svg|250px|thumb|right|The point ''x'', represented by a dot on the left of the picture, and the closed set ''F'', represented by a closed disk on the right of the picture, are separated by their neighbourhoods ''U'' and ''V'', represented by larger [[open disk]]s. 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.]]

A [[topological space]] ''X'' is a '''regular space''' if, given any [[closed set]] ''F'' and any [[Point (geometry)|point]] ''x'' that does not belong to ''F'', there exists a [[neighbourhood (topology)|neighbourhood]] ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are [[Disjoint sets|disjoint]]. Concisely put, it must be possible to [[separated set|separate]] ''x'' and ''F'' with disjoint neighborhoods.

A '''{{visible anchor|T3 space|text=T<sub>3</sub> space}}''' or '''{{visible anchor|regular Hausdorff space}}''' is a topological space that is both regular and a [[Hausdorff space]]. (A Hausdorff space or T<sub>2</sub> space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T<sub>3</sub> if and only if it is both regular and T<sub>0</sub>. (A T<sub>0</sub> 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 T<sub>0</sub>, and each T<sub>0</sub> 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 "T<sub>3</sub>" are not uncommon, there is significant variation in the literature: some authors switch the definitions of "regular" and "T<sub>3</sub>" as they are used here, or use both terms interchangeably. This article uses the term "regular" freely, but will usually say "regular Hausdorff", which is unambiguous, instead of the less precise "T<sub>3</sub>". For more on this issue, see [[History of the separation axioms]].

A {{em|{{visible anchor|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]].