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 (points are closed), Δ-Hausdorff, weak Hausdorff, KC space, k-Hausdorff, and Hausdorff (T₂); see the following for explanations.

A k-Hausdorff space 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 ${\displaystyle f:[0,1]\to X}$ is continuous then ${\displaystyle f([0,1])}$ is closed in ${\displaystyle X.}$ Every weak Hausdorff space is ${\displaystyle \Delta }$-Hausdorff, and every ${\displaystyle \Delta }$-Hausdorff space is a T₁ space. A space is Δ-generated if its topology is the finest topology such that each map ${\displaystyle f:\Delta ^{n}\to X}$ from a topological ${\displaystyle n}$-simplex ${\displaystyle \Delta ^{n}}$ to ${\displaystyle X}$ is continuous. ${\displaystyle \Delta }$-Hausdorff spaces are to ${\displaystyle \Delta }$-generated spaces as weak Hausdorff spaces are to compactly generated spaces.