Regular Paracompact Space is not necessarily Metrizable/Proof 1

Proof
Let $T$ be the right half-open interval space.

From Right Half-Open Interval Space is Regular, $T$ is a regular space.

From Right Half-Open Interval Space is Paracompact, $T$ is a paracompact space.

From Right Half-Open Interval Space is not Metrizable, $T$ is not a metrizable space.