# Cofibrations and mapping spaces in compactly generated weak Hausdorff spaces

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?

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.