 Compactly generated space

A compactly generated space, in the k-space convention used here, is a topological space whose closed subsets are detected by continuous maps from compact Hausdorff spaces. A subset $A\subseteq X$ is closed exactly when $f^{-1}(A)$ is closed for every continuous $f:K\to X$ with $K$ compact Hausdorff.

Every metric space has this property. If $A$ is not closed, choose $x_n\in A$ tending to $x\notin A$. The map from $K=\{0\}\cup \{1/n:n\ge 1\}$ sending $0$ to $x$ and $1/n$ to $x_n$ is continuous, but the inverse image of $A$ is not closed. A compact test therefore detects the failure.

The kification $kX$ imposes exactly these compact-test closed sets on the same underlying set. Compactly generated products use $k(X\times Y)$ so that products remain inside the chosen category. Some authors additionally require the weak Hausdorff property when saying compactly generated; that is an extra convention, used in the cited lecture notes.^[ref]

In topology, a **compactly generated space** is a type of topological space that can be characterized by its relationship with compact subsets. Specifically, a topological space \( X \) is said to be compactly generated if it is Hausdorff and a topology on \( X \) can be described in terms of its compact subsets.