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 be a set given the particular point topology with particular point The space is locally Hausdorff at since is an isolated point in and the singleton is a Hausdorff neighbourhood of For any other point any neighbourhood of it contains and therefore the space is not locally Hausdorff at

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

References