T3 Space with Sigma-Locally Finite Basis is T4 Space/Proof 2

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Theorem

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

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

Then:

$T$ is a $T_4$ space

Proof

From T3 Space with Sigma-Locally Finite Basis is Paracompact:

$T$ is a paracompact space

From $T_3$ Space is Fully $T_4$ iff Paracompact:

$T$ is a fully $T_4$ space.

From Fully $T_4$ Space is $T_4$ Space:

$T$ is a $T_4$ space.

$\blacksquare$