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From Wikipedia, the free encyclopedia
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.[1] 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.[2][3]
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) |
The notion was introduced by M. C. McCord[4] 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.
A k-Hausdorff space[5] is a topological space which satisfies any of the following equivalent conditions:
A Δ-Hausdorff space 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. A space is Δ-generated if its topology is the finest topology such that each map from a topological -simplex to is continuous. -Hausdorff spaces are to -generated spaces as weak Hausdorff spaces are to compactly generated spaces.
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