Regular Paracompact Space is not necessarily Metrizable/Proof 2

From ProofWiki
Jump to navigation Jump to search

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 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