Locally Hausdorff space

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 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".
* An infinite set in the finite complement topology is neither Hausdorff nor locally Hausdorff.
* If "G" is a topological group that is locally Hausdorff at "x" for some point "x" of "G", then "G" is locally Hausdorff at every point. 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"

ee also

* Hausdorff space


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