Show that \(\RR\) is separable.
Solution
1. The subset \(\mathbb Q\subset\mathbb R\) is countable.
The map \[ \mathbb Z\times\mathbb Z_{>0}\longrightarrow\mathbb Q, \qquad (p,q)\longmapsto p/q, \] is surjective, and \(\mathbb Z\times\mathbb Z_{>0}\) is countable.
2. The subset \(\mathbb Q\) is dense in \(\mathbb R\).
Let \((a,b)\) be a nonempty open interval. Choose \(q\in\mathbb Z_{>0}\) with \(q(b-a)>1\). Then there is an integer \(p\) with \[ qa<p<qb, \] so \(p/q\in(a,b)\cap\mathbb Q\). Hence every nonempty basic open set meets \(\mathbb Q\).
3. Therefore \(\mathbb R\) has a countable dense subset and is separable.