Hausdorff space
manifold with boundary

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

Further discussion of separated spaces may be found in the article Hausdorff space.

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

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

Embeddings into compact Hausdorff spaces may be of particular interest.

There are locally Hausdorff spaces where a sequence has more than one limit.

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

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

This can never happen for a Hausdorff space.

In particular, it is not a Hausdorff space.

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

These are what we normally call normal Hausdorff spaces.

Let be a Hausdorff space and a function.

For Hausdorff spaces, this concept is of course trivial.

A normal Hausdorff space is also called a T space.

This is to say, compact Hausdorff space is normal.

A space is called a continuum if it a compact, connected Hausdorff space.

The order topology makes X into a completely normal Hausdorff space.

The irreducible components of a Hausdorff space are just the singleton sets.

Assume that is a compact locally connected Hausdorff space.

However, unless k is a finite field no variety is ever a Hausdorff space.

In particular, a normal Hausdorff space is the same thing as a T space.

Every ultrafilter on a compact Hausdorff space converges to exactly one point.

Singleton points (and thus finite sets) are closed in Hausdorff spaces.

In fact, for Hausdorff spaces, paracompactness and full normality are equivalent.