Ian Morris quoted the following:

For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\subset C(X,\mathbb{R})$ such that $f(x)=\inf\{g(x)\colon g \in \mathcal{F}\}$ for every $x \in X$.

If $X$ is metrisable then $\mathcal{F}$ may be taken to be countable.

I am interested in the bold sentence: If $X$ is metrizable, then the class of continuous functions $\mathcal{F}$ can be countable.

Question: If we define a function $f:X \rightarrow [\infty,+\infty)$ such that for any real number $c \in \mathbb{R},$ its pre-image $f^{-1}(-\infty,c)$ is a $F_{\sigma}$ set, can we get a class $\mathcal{F}$ of real-valued upper semicontinuos functions on $X$ such that $f(x) = \inf \{ g(x):g \in \mathcal{F} \}$ and **$\mathcal{F}$ is countable**?