Lemma 3

Let X be a topological space whose topology possesses a countable basis. Every discrete subset of X is countable.

Let $\mathscr{B}$ be a countable basis of the topology of X and let D be a discrete subset of X. For each element $x$ of D, there exists an element $B_x$ of $\mathscr{B}$ such that $B_x\cap D =\{x\}$. The mapping from D into $\mathscr{B}$ defined by $x\mapsto B_x$ is injective, whence the lemma.