Locally Hausdorff space explained

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

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

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

Notes and References

1. .
2. Niefield . S. B. . A note on the locally Hausdorff property . Cahiers de topologie et géométrie différentielle . 1983 . 24 . 1 . 87–95 . 2681-2398., Lemma 3.2
3. Baillif . Mathieu . Gabard . Alexandre . Manifolds: Hausdorffness versus homogeneity . Proceedings of the American Mathematical Society . 2008 . 136 . 3 . 1105–1111 . 10.1090/S0002-9939-07-09100-9. free . math/0609098 ., Lemma 4.2