Subset of Sigma-Locally Finite Set of Subsets is Sigma-Locally Finite

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $\FF \subseteq \powerset S$ be a set of subsets of $S$.

Let $\GG \subseteq \FF$.

If $\FF$ is $\sigma$-locally finite then $\GG$ is $\sigma$-locally finite.

Proof
By definition of $\sigma$-locally finite:
 * $\FF = \ds \bigcup_{n \in \N} \FF_n$

where $\FF_n$ is locally finite for each $n \in \N$.

For each $n \in \N$, let:
 * $\GG_n = \GG \cap \FF_n$

We have:

From Intersection is Subset:
 * $\GG_n \subseteq \FF_n$

From Subset of Locally Finite Set of Subsets is Locally Finite:
 * $\GG_n$ is locally finite for each $n \in \N$

By definition, $\GG$ is $\sigma$-locally finite.