# separable space and open covers

If a topological space $X$ is separable, then every open cover of $X$ must be countable? since $X$ is separable , then there exists a countable dense subset $S$. This implies, in every open cover any set must intersect with $S$.

No, definitely not. The Mrówka space $\Psi$ is a separable space with an irreducible open cover of cardinality $2^\omega=\mathfrak{c}$. (Irreducible means that it has no proper subcover.) The Katětov extension of $\Bbb N$ is a separable space that has an irreducible open cover of cardinality $2^{2^\omega}=2^{\mathfrak{c}}$.
• @ghb: A countable set has $2^\omega$ different subsets. Moreover, two different open sets can have the same intersection with a dense subset. – Brian M. Scott Nov 29 '13 at 4:28