Let $\mathcal{C}$ be a collection of subsets of a topological space $X$

$\mathcal{C}$ is said to be locally finite if for all $x\in X$ there is a neighbourhood $U$ of $x$ such that $U \cap A = \varnothing$ for all but finitely many $A \in \mathcal{C}$

Similarly, $\mathcal{C}$ is said to be locally countable if for all $x\in X$ there is a neighbourhood $U$ of $x$ such that $U \cap A = \varnothing$ for all but countably many $A \in \mathcal{C}$