Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $\AA$ be a locally finite set of subsets.
Then:
- $\AA$ is a $\sigma$-locally finite set of subsets
Proof
For each $n \in \N$, let
- $\AA_n = \AA$.
Then:
- $\AA = \ds \bigcup_{n \in \N} \AA_n$
The result follows.
$\blacksquare$