Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$

Question

Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$, $$ x_n \to x \iff [f(x_n) \to f(x) \quad \forall f \in \mathcal F]. \quad (\star) $$

Let $\tau$ be the metric topology on $E$ induced by $d$ and $\tau'$ the initial topology induced by $\mathcal F$. Clearly, $\tau' \subset \tau$. It's possible that $\tau' \neq \tau$.

My question Are there some conditions on $\mathcal F$ that ensures $\tau' = \tau$?

Thank you so much for your elaboration!

Answer 1

If $\mathcal{F}$ is countable, $\tau'$ is metrizable with a compatible metric $\rho$ given by $$\rho(x,y)=\sum_n 2^{-n}~|f_n(x)-f_n(y)|\wedge1$$ for some enumeration of $\mathcal{F}$.

Since the topology of a metric space is determined by the convergent sequences, this is a sufficient condition.