locally finite collection

Let $𝒞$ be a collection of subsets of a topological space $X$.

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

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