Show that if \(X\) has a countable dense subset, every collection of disjoint open sets in \(X\) is countable.
Solution
Let \(D\subset X\) be countable and dense, and let \(\mathcal U\) be a collection of pairwise disjoint nonempty open sets. For each \(U\in\mathcal U\), density gives a point \(d_U\in D\cap U\). Since the members of \(\mathcal U\) are disjoint, the points \(d_U\) are distinct. Thus \[ U\longmapsto d_U \] is an injection \(\mathcal U\hookrightarrow D\). Hence \(\mathcal U\) is countable.