Local Finiteness

Definition

Consider a Topological Space $(X,\mathcal{T})$ and a collection of subsets $\mathcal{A}=\{A_i{\}}_{i\in I}$ of $X$. The collection $\mathcal{A}$ is locally finite in $X$ if every point $x$ of $X$ has a Neighborhood $U_x\in \mathcal{T}$ that intersects only finitely many elements of $\mathcal{A}$. $\forall x\in X,\exists {\mathcal{N}}_x\text{s.t.}|{\mathcal{N}}_x\cap \mathcal{A}|<\infty$ where ${\mathcal{N}}_x$ is a Neighborhood of $x$.