# Uncountably many disjoint closed sets

Is it true that if I can find in a topological space uncountably many pairwise disjoint nonempty closed sets, then the space is not separable? I know it is true for open sets (from ccc), but for closed too?

No, a point is a closed set in the usual topology on $[0,1]$ therefore $$[0,1]=\bigcup_{x\in[0,1]} \{x\}$$ provides a counterexample.