= Kification
{title2=$kX$}
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.<Compactly Generated Spaces (18.906 Lecture 2)>[Definitions 1.3–1.4 and section 2]{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.
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