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.

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