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