Countably Compact Metric Space is Sequentially Compact

Theorem

Let $M$ be a countably compact metric space.


Then $M$ is sequentially compact.


Proof

This follows directly from the results:

Metric Space is First-Countable
Countably Compact First-Countable Space is Sequentially 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 $5$: Metric Spaces
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.