Let $(X,\tau)$ be a compactly generated Hausdorff space, that is, a set $F\subset X$ is closed if and only if the intersection of $F$ with every compact set is closed.
Further let $\mathcal F$ be a family of continuous functions separating points from compact sets, that is, such that for every $x\in X$ and every compact $K$ there exists $f\in\mathcal F$ and $a=a_{x,K}>0$ with $f(x)=0$ and $f\geq a$ on $K$.
Question: Is it true that $\mathcal F$ generates $\tau$?
Comments:
- The statement above is supposed to mimick the analogous one for closed sets (in place of compact sets):
If $\mathcal F$ is a family of continuous functions separating points from closed sets, then $\mathcal F$ generates $\tau$
a proof of which can be found here.
The reason for assuming Hausdorffness is that (otherwise) there are many (in)equivalent definitions of compactly generated topological space (see here).
If necessary, I am happy to assume that $(X,\tau)$ is completely regular, so that $\mathcal F$ is known a priori to be generated by the family of all continuous functions.
