nLab weakly Hausdorff topological space

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.

The notion also also makes sense as stated for locales. Where their theory overlaps (in sober spaces and topological locales), the notions of weak Hausdorffness agree, given the ultrafilter theorem (which implies that all compact Hausdorff spaces/locales are sober/topological).

Properties

Weak Hausdorffification

(this is a left adjoint …) See for now (Strickland 2009, Proposition 2.22), for the weak Hausdorffification of a compactly-generated topological space. Note in particular that the construction of the weak Hausdorffification of a compactly-generated space is a quotient by a closed equivalence relation given by a one-step construction (the intersection of all closed equivalence relations on the space), as opposed to the equivalence relation generated by a possibly transfinite procedure as in the Hausdorffification of an arbitrary topological space.

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.

(Peter May, MO answer, April 2015)

Related concepts

• model structure on compactly generated weak Hausdorff spaces

References

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

• Michael C. McCord, Section 2 of: Classifying Spaces and Infinite Symmetric Products, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 (jstor:1995173, pdf)

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.

A survey is in

• Neil Strickland, The category of CGWH spaces (2009) (pdf)