Proposition 6

If $\mathfrak{B}$ is a base of the topology of a topological space $X$, there is a dense set $D$ in $X$ such that $\mathrm{Card}(D) \leq \mathrm{Card}(\mathfrak{B})$.

We may restrict ourselves to the case in which none of the sets of $\mathfrak{B}$ is empty (the non-empty sets of $\mathfrak{B}$ already form a base of the topology of $X$). For each $U \in \mathfrak{B}$, let $x_V$ be a point of $U$; it follows from Proposition 3 of no. 3 that the set $D$ of the points $x_U$ is dense in $X$, and we have $\mathrm{Card}(D) \leq \mathrm{Card}(\mathfrak{B})$ (Set Theory, Chapter III, § 3, no. 2, Proposition 3).