Sigma-local Finiteness

Definition

Consider a Topological Space $(X,\mathcal{T})$ and a collection of subsets $\mathcal{B}=\{B_i\subset X{\}}_{i\in I}$ of $X$. The collection $\mathcal{B}$ is $\sigma$-locally finite if $\mathcal{B}$ in the union of countable collection of Locally Finite families $\{{\mathcal{B}}_n{\}}_{n\in \mathbb{N}}$. $\exists \{{\mathcal{B}}_n{\}}_{n\in \mathbb{N}}\text{s.t.}\mathcal{B}={\bigcup }_{n\in \mathbb{N}}{\mathcal{B}}_n$