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]

The kification $kX$ of a topological space $X$ has the same points and the finest topology making every original continuous map $K\to X$ from a compact Hausdorff space continuous. Equivalently, $A\subseteq kX$ is closed exactly when $f^{-1}(A)$ is closed in $K$ for every such test map. This topology is at least as fine as the original one, so the identity $kX\to X$ is continuous.^[ref]

If $X$ is already a compactly generated space, its topology is recovered unchanged. In particular this holds for metric spaces. The operation is idempotent: $k(kX)=kX$. It lets one form products and mapping spaces within compactly generated spaces: the categorical product is the kification of the ordinary product. Compact test maps retain their original continuity, while the new topology supplies the closed sets they detect.