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](https://mathoverflow.net/a/36390/477203) that $\tau' \neq \tau$.



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


Thank you so much for your elaboration!



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


