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 Hausdorff under the subspace topology.

Here are some facts:

Every Hausdorff space is locally Hausdorff.
Every locally Hausdorff space is T₁.
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 that 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).

See also