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

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

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:

\(\ds \GG\)

\(=\)

\(\ds \GG \cap \FF\)

Intersection with Subset is Subset

\(\ds \)

\(=\)

\(\ds \GG \cap \paren{\bigcup_{n \in \N} \FF_n}\)

Definition of $\FF$

\(\ds \)

\(=\)

\(\ds \bigcup_{n \in \N} \GG \cap \FF_n\)

Intersection Distributes over Union

\(\ds \)

\(=\)

\(\ds \bigcup_{n \in \N} \GG_n\)

Definition of $\GG_n$

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.

$\blacksquare$