Regular Paracompact Space is not necessarily Metrizable/Proof 2

Proof
Let $T$ be the Sorgenfrey line.

From Sorgenfrey Line satisfies all Separation Axioms, $T$ is a regular space.

From Sorgenfrey Line is Paracompact, $T$ is a paracompact space.

From Sorgenfrey Line is not Metrizable, $T$ is not a metrizable space.