Locally Hausdorff space

Separation axioms
in topological spaces
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)
	History;

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.^[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 $G$ 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).

References

^ Niefield, Susan B. (1991), "Weak products over a locally Hausdorff locale", Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, pp. 298–305, doi:10.1007/BFb0084228, MR 1173020.
^ Niefield, S. B. (1983). "A note on the locally Hausdorff property". Cahiers de topologie et géométrie différentielle. 24 (1): 87–95. ISSN 2681-2398., Lemma 3.2
^ Baillif, Mathieu; Gabard, Alexandre (2008). "Manifolds: Hausdorffness versus homogeneity". Proceedings of the American Mathematical Society. 136 (3): 1105–1111. arXiv:math/0609098. doi:10.1090/S0002-9939-07-09100-9., Lemma 4.2
^ Niefield 1983, Proposition 3.4.
^ Niefield 1983, Proposition 3.5.