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.^[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)
History

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

1. 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.
2. 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
3. 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
4. Niefield 1983, Proposition 3.4.
5. Niefield 1983, Proposition 3.5.