This is something which is not clear to me. Take any countable subset $C$ of a compact set $K$ in a locally convex topological vector space $X$. Can we conclude that there is a point $x\in X$ such that $C\cup\{x\}$ is compact? I am mostly concerned with duals of Banach spaces equipped with the weak*-topology.
2026-03-31 14:24:21.1774967061
Countable subsets of TVSs
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Take compact $[0,1]$ and consider its rational points $Q$. You need to add uncountably many points to make $Q$ compact.