# nLab weakly Hausdorff topological space

Weakly Hausdorff spaces

# Weakly Hausdorff spaces

## Definition

A topological space $X$ is weakly Hausdorff (or weak Hausdorff) if for any compact Hausdorff space $K$ and every continuous map $f\colon K\to X$, the image $f(K)$ is closed. Every weakly Hausdorff space is $T_1$ (that is every point is closed), and every Hausdorff space is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used, the assumption of being weakly Hausdorff suffices. See also compactly generated space.

We have given the definition for topological spaces, but it also makes sense as stated for locales. Where these overlap (sober spaces and topological locales), they agree given the ultrafilter theorem (which implies that all compact Hausdorff spaces/locales are sober/topological).

## Properties

### Weak Hausdorffification

(this is a left adjoint …)

### Pushouts

Write CGWH for the category of compactly generated weakly Hausdorff topological spaces, and $CGH$ for compactly generated Hausdorff topological spaces. Both are convenient categories of topological spaces that both admit a homotopy hypothesis-comparison to simplicial sets, but CGWH has a key further property:

The construction of pushouts is better behaved in CGWH than in CGH. Specifically, CHWH is closed under pushouts, one leg of which is the inclusion of a closed subspace. CGH does not have such nice behavior, and pushouts like that are used all over The Geometry of Iterated Loop Spaces, specifically in the construction of a monad from an operad and in the use of geometric realizations of simplicial spaces.

## References

The category of compactly generated weakly Hausdorff topological spaces was introduced in

• M. C. McCord, (1969), Classifying spaces and infinite symmetric products, Transactions of the American Mathematical Society 146: 273–298, doi:10.2307/19951

as a more convenient setting than Steenrod’s compactly generated Hausdorff spaces, given that the latter is not closed under many colimits (for instance quotients) as computed in $Top$.

• André Henriques et al, Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?, MO discussion, 2010.

Last revised on April 5, 2021 at 22:11:26. See the history of this page for a list of all contributions to it.