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]
Separation axioms in topological spaces | |
---|---|
Kolmogorov classification | |
T0 | (Kolmogorov) |
T1 | (Fréchet) |
T2 | (Hausdorff) |
T2½ | (Urysohn) |
completely T2 | (completely Hausdorff) |
T3 | (regular Hausdorff) |
T3½ | (Tychonoff) |
T4 | (normal Hausdorff) |
T5 | (completely normal Hausdorff) |
T6 | (perfectly normal Hausdorff) |
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 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).