Kification ($kX$)

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.