Locally Hausdorff space

Definition
A topological space $$X$$ is termed locally Hausdorff if it satisfies the following equivalent conditions:


 * 1) For every point $$x \in X$$, there is an open subset $$U$$ of $$X$$ containing $$x$$ which is Hausdorff in the subspace topology.
 * 2) For every point $$x \in X$$, and every open subset $$V$$ of $$X$$ containing $$x$$, there is an open subset $$U$$ of $$X$$ contained in $$V$$, and which is Hausdorff in the subspace topology from $$X$$.
 * 3) $$X$$ is a union of open subsets each of which is a Hausdorff space with the subspace topology.
 * 4) $$X$$ has a basis comprising Hausdorff spaces.

Formalisms
Note that since Hausdorffness is hereditary, some variants of the locally operator all collapse to the same meaning. In particular, every point being contained in an open Hausdorff subset is equivalent to having a basis of open Hausdorff subsets.