Sigma-Locally Finite Cover has Locally Finite Refinement

Theorem

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

Let $\SS = \ds \bigcup_{n \mathop = 0}^\infty \SS_n$ be a $\sigma$-locally finite cover of $X$, where each $\SS_n$ is locally finite for all $n \in \N$.

Then:

there exists a locally finite refinement $\AA$ of $\SS$.

Proof

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

$B_n = \bigcup \SS_n$

Let:

$\BB = \set {B_n : n \in \N}$

Lemma 1

$\BB$ is a cover of $X$

$\Box$

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

$C_n = B_n \setminus \ds \bigcup_{i \mathop < n} B_i$

Let:

$\CC = \set {C_n : n \in \N}$

Lemma 2

$\CC$ is a locally finite refinement of $\BB$

$\Box$

From Sigma-Locally Finite Cover and Countable Locally Finite Cover have Common Locally Finite Refinement:

there exists a common locally finite refinement $\AA$ of both $\CC$ and $\SS$.

$\blacksquare$