Locally Hausdorff space
| 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 [math]\displaystyle{ X }[/math] be a set given the particular point topology with particular point [math]\displaystyle{ p. }[/math] The space [math]\displaystyle{ X }[/math] is locally Hausdorff at [math]\displaystyle{ p, }[/math] since [math]\displaystyle{ p }[/math] is an isolated point in [math]\displaystyle{ X }[/math] and the singleton [math]\displaystyle{ \{p\} }[/math] is a Hausdorff neighbourhood of [math]\displaystyle{ p. }[/math] For any other point [math]\displaystyle{ x, }[/math] any neighbourhood of it contains [math]\displaystyle{ p }[/math] and therefore the space is not locally Hausdorff at [math]\displaystyle{ x. }[/math]
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 [math]\displaystyle{ G }[/math] is a topological group that is locally Hausdorff at some point [math]\displaystyle{ x \in G, }[/math] then [math]\displaystyle{ G }[/math] is Hausdorff. This follows from the fact that if [math]\displaystyle{ y \in G, }[/math] there exists a homeomorphism from [math]\displaystyle{ G }[/math] to itself carrying [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y, }[/math] so [math]\displaystyle{ G }[/math] 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., 1488, Springer, Berlin, pp. 298–305, doi:10.1007/BFb0084228.
- ↑ Niefield, S. B. (1983). "A note on the locally Hausdorff property". Cahiers de topologie et géométrie différentielle 24 (1): 87–95. ISSN 2681-2398. http://www.numdam.org/item/?id=CTGDC_1983__24_1_87_0., Lemma 3.2
- ↑ Baillif, Mathieu; Gabard, Alexandre (2008). "Manifolds: Hausdorffness versus homogeneity". Proceedings of the American Mathematical Society 136 (3): 1105–1111. doi:10.1090/S0002-9939-07-09100-9., Lemma 4.2
- ↑ Niefield 1983, Proposition 3.4.
- ↑ Niefield 1983, Proposition 3.5.
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