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]

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 ${\displaystyle X}$ be a set given the particular point topology with particular point ${\displaystyle p.}$ The space ${\displaystyle X}$ is locally Hausdorff at ${\displaystyle p,}$ since ${\displaystyle p}$ is an isolated point in ${\displaystyle X}$ and the singleton ${\displaystyle \{p\}}$ is a Hausdorff neighbourhood of ${\displaystyle p.}$ For any other point ${\displaystyle x,}$ any neighbourhood of it contains ${\displaystyle p}$ and therefore the space is not locally Hausdorff at ${\displaystyle x.}$

Properties

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.^[3]

Every locally Hausdorff space is T₁.^[4] 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.^[5]

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).

References