$\mathcal{S}$ is a set with discrete elements. Is it a good practice to write $|\mathcal{S}| < \infty$ in order to say $\mathcal{S}$ is discrete? This is not a common notation, but is short and effective in my view.
Note: it is also not of infinitely many elements.
This doesn't actually make sense. See here for more information about why a set with discrete elements isn't an actual thing.
$|\mathcal S|<\infty$ means that $\mathcal S$ is finite, and tells you nothing about the topology on the set. It's perfectly fine to use this notation to denote finiteness.
