Regular space
A topological space is called regular if for any closed subset F and point x not belonging to F, there exist two disjoint open subsets U and V with x ∈ U and F ⊆ V. A space is called a T3-space if it is both regular and T0, or equivalently, both regular and Hausdorff. (In older literature, the meanings of the terms "regular" and "T3-space" were reversed — see Separation axioms. Wikipedia does not follow this older convention.) The regularity condition is an example of a separation axiom.
In regular spaces, we can separate points from closed sets with open neighborhoods; in Hausdorff spaces, we can separate points from other points with open neighborhoods. In general, neither of these conditions implies the other (since points need not be closed and since there typically exist closed sets which are not singletons). The T0 property says that all singletons are closed, so it is precisely the property which, together with regularity, implies Hausdorff.
Occasionly, one can deduce regularity from the Hausdorff property. For instance, if a Hausdorff space is locally compact, then it is regular.
See also: Completely regular space.