Regular Space with Sigma-Locally Finite Basis is Perfectly Normal Space

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Theorem

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

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

Then:

$T$ is a perfectly normal space

Proof

By definition of regular topological space:

$T$ is a $T_3$ space

$T$ is a $T_0$ (Kolmogorov) space

From T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space:

$T$ is a perfectly $T_4$ space

From Regular Space is $T_2$ Space:

$T$ is a $T_2$ space

From $T_2$ Space is $T_1$ Space:

$T$ is a $T_1$ space

By definition, $T$ is a perfectly normal space.

$\blacksquare$