1. The problem statement, all variables and given/known data 'In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space.' http://en.wikipedia.org/wiki/Separable_metric_space Let (X,d) be a metric space. If X is countable than it immediately satisfies being a separable metric space? Because just choose X itself as the subset. The closure of X must be X. Hence there exists a countable dense subset, namely X itself. 3. The attempt at a solution Is this correct? Or they referring to proper subsets only?