Compactly generated conventions for based homotopy

A space $X$ is weak Hausdorff (WH) if every continuous map $u:K\to X$ from a compact Hausdorff space has closed image. Here compactness and Hausdorffness are separate requirements.

A subset $F\subseteq X$ is k-closed if $u^{-1}(F)$ is closed for every such test $u$. The space $kX$ has the same underlying set as $X$ and these closed sets. A space is compactly generated (CG) if $kX=X$, and CGWH if it is both CG and WH. The topology and mapping properties are established in the following lemmas.

For CG spaces put $X{\times }_{k}Y=k(X\times Y)$. Let $C_0(Y,Z)$ be the continuous maps with subbasic opens $W(u,K,O)=\{f:Y\to Z\text{ continuous}:f(u(K))\subseteq O\},$ where $K$ is compact Hausdorff, $u:K\to Y$ is continuous, and $O$ is open in $Z$. Put $C(Y,Z)=k{C}_0(Y,Z)$. For based spaces, $C_{\ast }(Y,Z)$ is the kification of the subspace of basepoint-preserving maps. Put $\mathrm{\Omega }Z=C_{\ast }((I,\{0,1\}),(Z,z_0))$, meaning the kified subspace of maps whose two endpoint values are $z_0$.

Categorical products and based constructions below use CGWH spaces unless explicitly stated otherwise. Cubical homotopy classes are still defined for arbitrary topological spaces. Homotopies relative to a subspace have the meaning of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints. For WH domains the test-image mapping topology equals the compact-Hausdorff-subspace convention; no analogous identification with arbitrary compact subsets is asserted here.