I am looking for a proof that for any compact interval $I\subseteq \mathbb{R}$ and any Polish (=complete, separable, and metrizable) space $X$, the space of continuous curves $C(I,X)$ with the uniform metric $$d(f,g) = \sup_{t\in I}d(f(t),g(t))$$ is Polish.
I was able to prove that the resulting space will be complete. But I do not know what functions I should choose for the separability. I have the feeling that this may involve the Stone Weierstrass theorem, but that involves the other space $C(X,I)$.
I looked back in my notes and see that this is theorem 2.4.3 in "A course in Borel sets" by S.M. Srivastava. I may fill in more details about the construction when I understand it better.