Editing Regular space (section)

== Relationships to other separation axioms ==
A regular space is necessarily also [[preregular space|preregular]], i.e., any two [[topologically distinguishable]] points can be separated by neighbourhoods.
Since a Hausdorff space is the same as a preregular [[Kolmogorov space|T<sub>0</sub> space]], a regular space which is also T<sub>0</sub> must be Hausdorff (and thus T<sub>3</sub>).
In fact, a regular Hausdorff space satisfies the slightly stronger condition [[Urysohn and completely Hausdorff spaces|T<sub>2½</sub>]].
(However, such a space need not be [[Completely Hausdorff space|completely Hausdorff]].)
Thus, the definition of T<sub>3</sub> may cite T<sub>0</sub>, [[T1 space|T<sub>1</sub>]], or T<sub>2½</sub> instead of T<sub>2</sub> (Hausdorffness); all are equivalent in the context of regular spaces.

Speaking more theoretically, the conditions of regularity and T<sub>3</sub>-ness are related by [[Kolmogorov quotient]]s.
A space is regular if and only if its Kolmogorov quotient is T<sub>3</sub>; and, as mentioned, a space is T<sub>3</sub> if and only if it's both regular and T<sub>0</sub>.
Thus a regular space encountered in practice can usually be assumed to be T<sub>3</sub>, 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 [[normal space|normality]], [[Pseudonormal space|pseudonormality]], [[paracompactness]], or [[local compactness]]) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.<ref>{{cite web |title=general topology - Preregular and locally compact implies regular |url=https://math.stackexchange.com/q/1272957 |website=Mathematics Stack Exchange}}</ref>
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 space|completely regular]], which is a stronger condition.
Regular spaces should also be contrasted with [[normal space]]s.