T3 Space with Sigma-Locally Finite Basis is Paracompact

Theorem

Let $T = \struct {S, \tau}$ be a $T_3$ topological space.

Let $\BB$ be a $\sigma$-locally finite basis of $T$.

Then:

$T$ is a paracompact

Proof

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

Let $\VV = \set{B \in \BB : \exists U \in \UU : B \subseteq U}$

Hence $\VV \subseteq \BB$.

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

$\VV$ is $\sigma$-locally finite

Let $x \in S$.

By definition of open cover:

$\exists U \in \UU : x \in U$

By definition of basis:

$\exists B \in \BB : x \in B \subseteq U$

Hence:

$B \in \VV$

It follows that $\VV$ is an open cover by definition.

By definition, $\VV$ is an open refinement of $\UU$.

It has been shown that:

every open cover of $T$ has an open $\sigma$-locally finite refinement

From Characterization of Paracompactness in T3 Space:

$T$ is paracompact

$\blacksquare$