Topological properties determined by closures of countable sets

We establish that a locally convex space must be metrizable if it has a Čech-complete dense subspace and show that there are locally convex spaces L of arbitrarily large extent such that $\bar{A}$ is σ-compact for any countable set $A\subset L$. Under Jensen's Axiom (⋄), we give an example of a metrizable space M which does not have a dense Čech-complete subspace while $\bar{A}$ is Čech-complete for every countable set $A\subset M$. Our results give a consistent answer to two published open questions.