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.
[1] A space is locally Hausdorff exactly if it can be written as a union of Hausdorff open subspaces.
[2] And in a locally Hausdorff space each point belongs to some Hausdorff dense open subspace.
For example, an infinite set with the cofinite topology is a T1 space that is not locally Hausdorff.
Every locally Hausdorff space is sober.
is a topological group that is locally Hausdorff at some point