Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1

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Theorem

Let $T = \struct{X, \tau}$ be a topological Space.

Let $\SS = \ds \bigcup_{n = 0}^\infty \SS_n$ be a cover of $X$, where each $\SS_n$ is a set of subsets of $X$ for each $n \in \N$.

For each $n \in \N$, let:

$B_n = \ds \bigcup \SS_n$

Let:

$\BB = \set{B_n : n \in \N}$

Then:

$\BB$ is a cover of $X$

Proof

Let $x \in X$.

By definition of cover of set:

$\exists S \in \SS : x \in S$

By definition of union:

$\exists n \in \N : S \in \SS_n$

From Set is Subset of Union:

$S \subseteq B_n$

By definition of subset:

$x \in B_n$

It follows by definition, $\BB$ is a cover of $X$.

$\blacksquare$