Regular Paracompact Space is not necessarily Metrizable/Proof 2

Proof
Let $T$ be the radial interval topology.

From Radial Interval Topology is Completely Normal, $T$ is a completely normal space.

Hence from Sequence of Implications of Separation Axioms, $T$ is a regular space.

From Radial Interval Topology is Paracompact, $T$ is a paracompact space.

From Radial Interval Topology is not Metrizable, $T$ is not a metrizable space.