Regular Paracompact Space is not necessarily Metrizable

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space which is regular and paracompact.

Then it is not necessarily the case that $T$ is metrizable.

Proof
Let $T$ be the Sorgenfrey line.

From Sorgenfrey Line is Regular, $T$ 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.