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)

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 be a set given the particular point topology with particular point The space is locally Hausdorff at since is an isolated point in and the singleton is a Hausdorff neighbourhood of For any other point any neighbourhood of it contains and therefore the space is not locally Hausdorff at

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 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).

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.
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