Regular Paracompact Space is not necessarily Metrizable/Proof 1
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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
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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metrizability