A regular space is a topological space (or variation) that has, in a certain sense, enough regular open subsets. The condition of regularity is one the separation axioms satsified by every metric space (in this case, by every pseudometric space).
Fix a topological space .
The classical definition is this: that if a point and a closed set are disjoint, then they are separated by neighbourhoods. In detail, this means:
Given any point and closed set , if , then there exist a neighbourhood of and a neighbourhood of such that is empty.
In many contexts, it is more helpful to change perspective, from a closed set that does not belong to, to an open set that does belong to. Then the definition reads:
Given any point and neighbourhood of , there exist a neighbourhood of and an open set such that but .
If we apply the regularity condition twice, then we get what at first might appear to be a stronger result:
Given any point and neighbourhood of , there exist a neighbourhood of and an open set such that but (where indicates topological closure).
Find and as above. Now apply the regularity axiom to and the interior of to get (and ).
In terms of the classical language of separation axioms, this says that and are separated by closed neighbourhoods.
Sometimes one includes in the definition that a regular space must be :
A space is if, given any two points, if each neighbourhood of either is a neighbourhood of the other, then they are equal.
Other authors use the weaker definition above but call a regular space a space, but then that term is also used for a (merely) regular space. An unambiguous term for the weaker condition is an space, but hardly anybody uses that.
Every space is Hausdorff.
Suppose every neighbourhood of meets every neighbourhood of ; by (and symmetry), it's enough to show that each neighbourhood of is a neighbourhood of . Use regularity to get and . Then cannot be a neighbourhood of , so is.
Since every Hausdorff space is , a less ambiguous term for a space is a regular Hausdorff space.
It is possible to describe the regularity condition fairly simply entirely in terms of the algebra of open sets. First notice the relevance above of the condition that ; we write in that case and say that is well inside . We now rewrite this condition in terms of open sets and regularity in terms of this condition.
Given sets , then iff there exists an open set such that but . Then is regular iff, given any open set , is the union of all of the open sets that are well inside .
This definition is suitable for locales. As the definition of a Hausdroff locale is rather more complicated, one often speaks of compact regular locales where classically one would have spoken of compact Hausdorff spaces. (The theorem that compact regular spaces and compact Hausdorff spaces are the same works also for locales, and every locale is , so compact regular locales and compact Hausdorff locales are the same.)
The condition that a space be regular is related to the regular open sets in , that is those open sets such that is the interior of its own closure. (In the Heyting algebra of open subsets of , this means precisely that is its own double negation; this immediately generalises the concept to locales.) Basically, we start with a neighbourhood of and reduce that to a closed neighbourhood of ; then is a regular open set.
This is enough to characterise regular spaces, as follows:
Given a neighbourhood of , there is a closed neighbourhood of that is contained in . Equivalently, has a regular open neighbourhood contained in . In other words, the closed neighbourhoods of , or equivalently the regular open neighbourhoods of , form a local base (a base of the neighbourhood filter) at .
In constructive mathematics, Definition 2 is good; then everything else follows without change, except for the equivalence with 1. Even then, the classical separation axioms hold for a regular space; they just are not sufficient.
Definition 5 suggests a slightly weaker condition, that of a semiregular space:
The regular open sets form a basis for the topology of .
As we've seen above, a regular space () is Hausdorff (); we can also remove the condition from the latter to get :
Given points and , if every neighbourhood of meets every neighbourhood of , then every neighbourhood of is a neighbourhood of .
A bit stronger than regularity is complete regularity; a bit stronger than is . The difference here is that we require that and be separated by a function, that is by a continuous real-valued function. See Tychonoff space for more.
A uniform space is automatically regular and even completely regular, at least in classical mathematics. In constructive mathematics this may not be true, and there is an intermediate notion of interest called uniform regularity.
Every regular space comes with a naturally defined (point-point) apartness relation: we say if there is an open set containing but not . This can be defined for any topological space and is obviously irreflexive, but in a regular space it is symmetric and a comparison, hence an apartness. For symmetry, if and , let be an open set containing and an open set such that and ; then (since ) while (since ). With the same notation, to prove comparison, for any we have either , in which case , or , in which case . Note that this argument is valid constructively; indeed, classically, the much weaker separation axiom is enough to make this relation symmetric, and it is a comparison on any topological space whatsoever.
Note that if a space is localically strongly Hausdorff (a weaker condition than regularity), then it has an apartness relation defined by if there are disjoint open sets containing and . If is regular, then this coincides with the above-defined apartness.