Regular Paracompact Space is not necessarily Metrizable

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Theorem

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


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


Proof 1

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.

$\blacksquare$


Proof 2

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.

$\blacksquare$


Sources