Weak Hausdorff space
| Separation axioms in topological spaces | |
|---|---|
| Kolmogorov classification | |
| T0 | (Kolmogorov) |
| T1 | (Fréchet) |
| T2 | (Hausdorff) |
| T2½ | (Urysohn) |
| completely T2 | (completely Hausdorff) |
| T3 | (regular Hausdorff) |
| T3½ | (Tychonoff) |
| T4 | (normal Hausdorff) |
| T5 | (completely normal Hausdorff) |
| T6 | (perfectly normal Hausdorff) |
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: every single-point set is closed.
- Δ-Hausdorff: the image of every path is closed.
- weak Hausdorff: the image under a continuous map of a compact Hausdorff space is closed.
- KC space: every compact subset is closed.
- k-Hausdorff: every compact subspace is Hausdorff.
- Hausdorff (T2): distinct points have disjoint neighborhoods.
These are further described in the below.