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)
T(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
 Hausdorff)
T6 (perfectly normal
 Hausdorff)

Examples and sufficient conditions

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  • 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

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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

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  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.