This talk’s purpose is to present some results on whether a topological space $X$ has a property $\mathcal P$ given that $\overline A$ has $\mathcal P$ for any countable set $A\subset X$. The respective line of research was outlined in a recent paper of A. Dow and the author. We prove, among other things, that there is a consistent example of a metric space $X$ such that $\overline A$ is \v Cech-complete for any countable $A\subset X$ but $X$ has no dense \v Cech-complete subspace. This talk will also feature some related results on general locally convex spaces.