I'm trying to solve Exercise 19, from Dugundji's Topology, pág. 92:
Let $X$ be a topological space in which every countable subset is closed. Is $X$ necessarily discrete?
I think the claim is false, but I couldn't figure out any counterexample.
2026-04-08 04:19:09.1775621949
If every countable subset of a space is closed, is the topology discrete?
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Let's try an example. We'd better take an uncountable set; how about $\Bbb R$. Let's make all countable subsets closed, and make $\Bbb R$ closed. Now is there a topology on $\Bbb R$ making all these sets and no others closed? And is it a discrete topology?