Existence of Paracompact Space which is not Compact

Theorem

There exists at least one example of a paracompact topological space which is not also a compact topological space.


Proof

Let $T = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.


From Real Number Line is Paracompact, $T$ is a paracompact space.

From Real Number Line is not Countably Compact, $T$ is not a countably compact space.

From Compact Space is Countably Compact, it follows that $T$ is not a compact space.

Hence the result.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness
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