 Compactly generated product

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]