Are WOT/SOT topologies hereditarily separable? Just out of curiosity, > Are weak and strong operator topologies on $B(H)$ hereditarily separable? In other words, if $S$ is a subset of $B(H)$, where $H$ is a separable Hilbert space, is $S$ separable in the relative WOT/SOT topology?

Yes.

Ernest Michael shows in On a theorem of Rudin and Klee, Proc. Amer. Math. Soc. 12 (1961), 921 that the space $Y^X$ of continuous functions $X\to Y$ between second countable spaces is hereditarily separable in the topology of pointwise convergence and also in the compact-open topology. His argument is beautiful and short -- the article is half a page long.

Since the strong operator topology on $B(H)$ is the subspace topology induced by $H^H$ with the topology of pointwise convergence, it follows that $B(H)$ with the strong operator topology is hereditarily separable. Since the weak operator topology is weaker, it is also hereditarily separable.