Exercise 10, § 1

Let $X$ be a topological space having a countable basis of open sets $\mathfrak{B}$. If $\mathfrak{U}$ is an uncountable set of nonempty open sets of $X$, there exists a set $V \neq \varnothing$ in $\mathfrak{B}$ contained in an uncountable infinity of sets of $\mathfrak{U}$.