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. 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 of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the . 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 , if is closed in for each compact Each compact subspace is closed and strongly locally compact. A space is if for each and each (not necessarily open) neighborhood of there exists a compact neighborhood of such that 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. A space is Hausdorff-compactly generated weak Hausdorff if and only if it is Hausdorff-compactly generated k-Hausdorff. 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. A is a topological space where the image of every path is closed; that is, if whenever is continuous then is closed in Every weak Hausdorff space is -Hausdorff, and every -Hausdorff space is a T1 space.