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.Template:Sfn 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.Template:Sfn

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).