Definition 4.18.1 (Locally Finite). Let $X$ be a topological space and $\mathcal{U}\subset 2^{X}$, then $\mathcal{U}$ is locally finite if for every $x \in X$, there exists $V \in \cn(x)$ such that $\bracs{U \in \mathcal{U}| V \cap U \ne \emptyset}$ is finite.