Is the space of simple predictable processes topologized by uniform convergence metrizable?

Question

Im following the book by Protter on Stochastic Integration and he is currently introducing Semimartingales. In doing so he first introduces the class of simple predictable processes $S$ and then topologizes them with uniform convergence. As I understand it, this topology defines the closed sets as being those that are closed under uniform convergence, meaning that a set $C$ is closed iff for every convergent subsequence $((X_t^n)_{t \in [0,\infty)})_{n \in \mathbb{N}} $ with $\lim_{n \to \infty} \sup_t |X^n_t-X_t| = 0$ and $X \in S$ one has $X \in C$. My topology background is not super strong so I was wondering, is this space metrizable?

Answer 1

Protter's topology on $\mathcal{S}$ is induced by the norm (see p. 52) $$ \left\|\|X-Y\|_{\infty}\right\|_{L^\infty} $$ where the $L^\infty$-norm indicates an essential supremum over $\omega\in\Omega$ and the $\infty$-norm is the sup-norm over $t\in[0,\infty)$. Therefore, $\mathcal{S}$ is even a normed space, so in particular metrizable.