We need to prove that any basis for this topology is uncountable. Def. Basis. Let (,) be a topological space. Theorem. Basis for a Topology 4 4. You will find one in a proof somewhere in this post! This topology consists of all unions of members of \(\mathcal{B}\). Prove that S is closed. Prove that \(U\) is open if \(\forall x\in U\), there exists \(B\in\mathcal{B}\) such that \(x\in B\subseteq U\). Hence the class of finite subset of \(\mathbb{Z}\) is countable. Definition. Otherwise, \(\mathcal{B}\) is not a basis for any topology. The collection of all open intervals \((a, b)\) whereas \(a,b\in\mathbb{Q}\) is the basis of \(\mathbb{R}\). Let \((X, \tau)\) be a topological space. Let \(S=\{0, 1, 1/2, 1/3, \dots, 1/n,\dots\}\). ▶ Proof. \(\mathcal{B}=\{(a,b): a,b\in\mathbb{Q}, a