I am reading through the book of Kosorok on Empirical Processes and I got stuck on a statement that seems to be clear to the author. To me, it is not clear at all.
In chapter 6.1 page 87 he states that the subspace $UC(T,\rho)$ of $l^{\infty}(T)$ is a Polish space. In this case, $l^{\infty}(T)$ is equipped with the supremum norm and $\rho$ is a semimetric on the arbitrary set $T$ and $UC(T,\rho):=\{ f: T\mapsto \mathbb{R} \mid f\ \text{is bounded and uniformly $\rho- $continuous}\}$.
I understand,why the space is complete w.r.t. the uniform metric, but not why it is separable. It is also not clear to me if this statement refers to the case of $T$ being compact. In neither case I am able to deduce the separability of $UC(T,\rho)$.
Any help is greatly appreciated!
If $T$ is an arbitrary set, $\ell^\infty(T)$ is not separable. Taking $T=\mathbb{N}$ already fails this. And if we take $\rho$ the discrete metric, $UC(T,\rho) = \ell^\infty(T)$ is also not separable.