Follow up: Density Characterized by Weak Topology

Question

This question is a follow-up on: this question.

Let $F$ be a non-empty subset of $C(X,Y)$, where $X,Y$ are Hausdorff (and for simplicity assume that $Y$ is metric). Let $\tau$ be the weak topology on $X$ generated by $F$. Let us denote $(X,\tau)$ by $\tilde{X}$ to avoid confusion.

• Is $F$ is dense in $C(\tilde{X},Y)$?

If this is indeed the case then it seems to me that $F$ is dense in $C(X,Y)$ if and only if $\tau$ generates the "original" topology on $X$.

Answer 1

No. For instance, let $X=Y$ and let $F$ consist of just the identity map. Then $\tau$ is just the original topology on $X$, but $F$ is not dense in $C(X,Y)$ in most cases.