T3 Space with Sigma-Locally Finite Basis is Paracompact
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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
$\blacksquare$