Weak Hausdorff space
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. 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].
Their strictness as separation properties in increasing order is T1,,,,, and ; see the following for explanations.
k-Hausdorff spaces
A ' is a topological space which satisfies any of the following equivalent conditions:- Each compact subspace is Hausdorff.
- The diagonal is k-closed in
- * A subset is '
Properties
- A k-Hausdorff space is weak Hausdorff. For if is k-Hausdorff and is a continuous map from a compact space then is compact, hence Hausdorff, hence closed.
- A Hausdorff space is k-Hausdorff. For a space is Hausdorff if and only if the diagonal is closed in and each closed subset is a k-closed set.
- A k-Hausdorff space is KC. A is a topological space in which every compact subspace is closed.
- To show that the coherent topology induced by compact Hausdorff subspaces preserves the compact Hausdorff subspaces and their subspace topology requires that the space be k-Hausdorff; weak Hausdorff is not enough. Hence k-Hausdorff can be seen as the more fundamental definition.
Δ-Hausdorff spaces