Editing Regular space (section)

== 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 [[interior (topology)|interior]]s of these closed neighbourhoods, we see that the [[regular open set]]s form a [[base (topology)|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 space|semiregular]]''.