I am reading these notes https://mkosorok.web.unc.edu/wp-content/uploads/sites/14747/2017/07/lecture06.pdf on empirical processes. I am confused by the statement on slide 38:
Note that $\ell^\infty(T)$ is separable if and only if $T$ is countable.
That is not true, is it? The notes define $\ell^\infty(T)$ as the space of all real-valued functions $f:T\rightarrow\mathbb R$ with $\Vert f\Vert_T := \sup_{t\in T}\vert f(t)\vert < \infty$. Consider $T=\mathbb N$, which is countable. There are many posts here on MathSE discussing the fact that $\ell^\infty(\mathbb N)$ is not separable. The notes discuss advanced results from functional analysis - I can't believe such a basic result would be misstated... or am I missing something here?