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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]

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.

For CW pairs $(X,A)$ and $(Y,B)$, write their product pair asUse the compactly generated product, equivalently the weak CW complex topology on products of cells. For arbitrary infinite CW complexes this can be finer than the ordinary product topology. The ordinary product topology already agrees when one factor is locally compact, or when both factors have countably many cells; in particular there is no distinction for finite CW complexes.^[ref]

The quotient of this pair is $(X/A)\wedge (Y/B)$. Setwise, the collapsed subset is exactly the set of pairs with at least one coordinate in its collapsed subspace; the compatible quotient constructions give the homeomorphism in the compactly generated space convention. In a multiplicative generalized cohomology theory, external products of relative classes of degrees $k,m$ lie in degree $k+m$ for this pair. Pullback along a diagonal map gives the relative cup product.