Assume that $X$ and $Y$ are compactly generated weak Hausdorff spaces (CGWH spaces for short). Assume that they are also well-pointed (so the inclusions of the base points are Hurewicz cofibrations).
Is then the mapping space $Y^X$ of pointed maps from $X$ to $Y$ again well-pointed (with base point the constant map from $X$ to the basepoint of $Y$)?


More generally, if $A\to X$ and $B\to Y$ are (unpointed) Hurewicz cofibrations in CGWH,
are the natural maps $B^X\to Y^X$ and $B^X\to (Y,B)^{(X,A)}$
cofibrations as well?


This seems to hold if $X$ is compact. Are there more general results?



Cofibrations and mapping spaces in compactly generated weak Hausdorff spaces
If the inclusion $A\to X$ is a cofibration, then $A$ is a $G\_\delta$ subset of $X$ (i.e. the intersection of a countable family of open subsets), as one can deduce easily from the fact that $\{0\}$ is $G\_\delta$ in $[0,1]$. If $T$ is uncountable and discrete then the basepoint in $[0,1]^T$ is not $G\_\delta$. Thus, the functor $(-)^T$ does not send the cofibration $\{0\}\to [0,1]$ to a cofibration.


