In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular, every Hausdorff space is weak Hausdorff. As a separation property, it is stronger than T1, which is equivalent to the statement that points are closed. Specifically, every weak Hausdorff space is a T1 space. The notion was introduced by M. C. McCord to remedy an inconvenience of working with the category of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the category of compactly generated weak Hausdorff spaces.

k-Hausdorff spaces

A **** is**** a topological space which satisfies any of**** the following equivalent**** conditions****:****

Properties

Δ-Hausdorff spaces

A **** is**** a topological space where the image of**** every path**** is**** closed****;**** that**** is****,**** if**** whenever**** is**** continuous**** then**** f([0, 1]) is**** closed**** in**** X.**** Every weak**** Hausdorff space is**** *Del**ta**-Hau**sd**or**ff**,*** and every *Del**ta**-Hau**sd**or**ff** spa**ce** is*** a T1**** space.**** A space is**** **** if**** its topology**** is**** the finest**** topology**** such**** that**** each**** map from**** a topological n-simplex *Del**ta**^n to*** X is**** continuous****.**** \Delta-Hausdorff spaces are to \Delta-generated spaces as weak Hausdorff spaces are to compactly generated spaces.