On which spaces can lower semicontinuous functions be approximated from below by bounded continuous functions?

Question

On metric spaces this can obviously be done by defining $F_n$ as in this question and then e.g setting $G_n(x) = \min \{ F_n ( x ), n \}$.

But what if the space in question is not metrisable? In my case I have a nuclear space, but I suspect that this should be possible on a much broader class of spaces.

Answer 1

This question has a clear answer in enter link description here. The correct category is that of perfectly normal spaces.