Characterization of Paracompactness in T3 Space/Statement 6 implies Statement 2

Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ Space.

If every open cover of $T$ has an open $\sigma$-locally finite refinement then:
 * every open cover of $T$ has a locally finite refinement

Proof
Let every open cover of $T$ have an open $\sigma$-locally finite refinement.

Let $\UU$ be an open cover of $T$.

Let $\VV$ be an open $\sigma$-locally finite refinement of $\UU$.

By definition of open $\sigma$-locally finite:
 * $\VV = \ds \bigcup_{n = 0}^\infty \VV_n$

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

For each $n \in \N$, let:
 * $W_n = \ds \bigcup \VV_n$

Let:
 * $\WW = \set{W_n : n \in \N}$

Lemma n

 * $\WW$ is a cover of $X$

For each $n \in \N$, let:
 * $A_n = W_n \setminus \ds \bigcup_{i < n} W_i$

Let:
 * $\AA = \set{A_n : n \in \N}$

Lemma n + 1

 * $\AA$ is a a locally finite refinement of $\WW$

Let:
 * $\BB = \set{A_n \cap V : n \in \N, V \in \VV_n}$

Lemma n + 2

 * $\BB$ is a locally finite refinement of $\VV$

From User:Leigh.Samphier/Topology/Refinement of a Refinement is Refinement of Cover:
 * $\BB$ is a locally finite refinement of $\UU$

Since $\UU$ was arbitrary, it follows that:
 * every open cover of $T$ has a locally finite refinement