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The compactly generated product of compactly generated spaces $X,Y$ iswhere the right-hand product has the ordinary product topology and $k$ denotes kification. Kification retains the same points but declares a subset closed when its inverse image is closed under every continuous map from a compact Hausdorff space. Its topology can be finer than the ordinary product topology.

For a compactly generated source $Z$, a set map $Z\to X{\times }_{k}Y$ is continuous exactly when both coordinate maps are continuous. Thus this is the categorical product in the compactly generated category, even when the ordinary topological product fails to be compactly generated. If one factor is locally compact Hausdorff, the ordinary product is already suitable.

For example, the compactly generated product of finite CW complexes has the usual product topology. The modification matters in general function-space and homotopy constructions, where products and mapping spaces must interact correctly.^[ref]

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.

For pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, the pointed mapping space ${\mathrm{Map}}_{\ast }(X,Y)$ consists of continuous maps $f:X\to Y$ with $f(x_0)=y_0$. It carries the subspace topology from the compact-open topology on the ordinary mapping space, commonly followed by kification when working with compactly generated spaces. Its basepoint is the constant map with value $y_0$.

For the pointed circle, ${\mathrm{Map}}_{\ast }(S^1,Y)$ is the loop space $\Omega Y$: a circle map is a path returning to its starting point. More generally, for well-behaved pointed spaces, path components of ${\mathrm{Map}}_{\ast }(S^n,Y)$ correspond to ${\pi }_n(Y,y_0)$. Thus a mapping space turns families of maps and homotopies into points and paths of another topological space. The choice of mapping-space topology ensures that this correspondence works continuously.