Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, comapctly gernerated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces.

There is the CG-ification $X_{CG}$ of a space $X$ and the WH-ification $X_{WH}$ of a compactly generated space $X$. These provide adjoint pairs between $Top$ and $CG$ and also between $CG$ and $CGWH$. Combined, they give my the CGWH-fication $(X_{CG})_{WH}=X_{CGWH}$ of a space $X$. The latter one is the one used implicitly in most of todays algebraic topology.

I'm interested in the question which of these constructions can change the (weak) homotopy type and which of them preserves it. It seems to me that at least in many cases, the weak homotopy type must be preserved. I really don't want to have $\Omega X$ to have a different homotopy type depending on whether I used the compactly generated topology on the mapping space or not.