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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?

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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.

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  • $\begingroup$ Thanks for this very clear and simple counterexample. $\endgroup$ – Sebastian Goette Dec 11 '19 at 20:08

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