Locally Hausdorff space

In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology.

Examples and sufficient conditions

 * Every Hausdorff space is locally Hausdorff.
 * There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
 * The line with two origins is locally Hausdorff (it is in fact locally metrizable) but not Hausdorff.
 * The etale space for the sheaf of differentiable functions on a differential manifold is not Hausdorff, but it is locally Hausdorff.
 * Let $$X$$ be a set given the particular point topology with particular point $$p.$$ The space $$X$$ is locally Hausdorff at $$p,$$ since $$p$$ is an isolated point in $$X$$ and the singleton $$\{p\}$$ is a Hausdorff neighbourhood of $$p.$$  For any other point $$x,$$ any neighbourhood of it contains $$p$$ and therefore the space is not locally Hausdorff at $$x.$$

Properties
A space is locally Hausdorff exactly if it can be written as a union of Hausdorff open subspaces. And in a locally Hausdorff space each point belongs to some Hausdorff dense open subspace.

Every locally Hausdorff space is T1. The converse is not true in general. For example, an infinite set with the cofinite topology is a T1 space that is not locally Hausdorff.

Every locally Hausdorff space is sober.

If $$G$$ is a topological group that is locally Hausdorff at some point $$x \in G,$$ then $$G$$ is Hausdorff. This follows from the fact that if $$y \in G,$$ there exists a homeomorphism from $$G$$ to itself carrying $$x$$ to $$y,$$ so $$G$$ is locally Hausdorff at every point, and is therefore T1 (and T1 topological groups are Hausdorff).