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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?

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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.

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It's possible that this is a misremembering of the following true statement: a first SECOND-countable space cannot have an uncountable discrete subspace.

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  • $\begingroup$ Of course it can: an uncountable discrete metric space is a counterexample (there are many others). Maybe you mean "second countable" (= having a countable base), in that case it would be true. $\endgroup$ – Henno Brandsma Jun 26 '13 at 11:19
  • $\begingroup$ @HennoBrandsma this is why i should never use the word "misremembering" $\endgroup$ – citedcorpse Jun 26 '13 at 11:22

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