Locally Hausdorff space

In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has an open neighbourhood that is a Hausdorff space under the subspace topology.[1]

Here are some facts:

• Every Hausdorff space is locally Hausdorff.
• Every locally Hausdorff space is T₁.[2]
• There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
• The bug-eyed line 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.
• A T₁ space need not be locally Hausdorff; an example of this is an infinite set given the cofinite topology.
• Let X be a set given the particular point topology. Then X is locally Hausdorff at precisely one point. From the last example, it will follow that a set (with more than one point) given the particular point topology is not a topological group. Note that if x is the 'particular point' of X, and y is distinct from x, then any set containing y that doesn't also contain x inherits the discrete topology and is therefore Hausdorff. However, no neighbourhood of y is actually Hausdorff so that the space cannot be locally Hausdorff at y.
• If G is a topological group that is locally Hausdorff at x for some point x of G, then G is Hausdorff. This follows from the fact that if y is a point of 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

1. 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, MR 1173020.
2. Clark, Lisa Orloff; an Huef, Astrid; Raeburn, Iain (2013), "The equivalence relations of local homeomorphisms and Fell algebras", New York Journal of Mathematics, 19: 367–394, MR 3084709. See remarks prior to Lemma 3.2.