Another nice property of Hausdorff spaces is that compact sets are always closed.

The definition of a Hausdorff space says that points can be separated by neighborhoods.

A paracompact subset of a Hausdorff space need not be closed.

In the following we will assume all groups are Hausdorff spaces.

These are what we normally call normal Hausdorff spaces.

Moreover, in a Hausdorff space, there is at most one limit to every filter base.

This can never happen for a Hausdorff space.

Embeddings into compact Hausdorff spaces may be of particular interest.

Every compact connected Hausdorff space, with more than one point, has at least two noncut-points.

In particular, it is not a Hausdorff space.