Paracompact Countably Compact Space is Compact

Theorem

Let $T = \struct {S, \tau}$ be a countably compact space which is also paracompact.

Then $T$ is compact.


Proof

From the definition of paracompact space, a paracompact space is also a metacompact space.

The result follows from Metacompact Countably Compact Space is Compact.

$\blacksquare$


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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