First-Countable Space is Sequentially Compact iff Countably Compact
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Theorem
A first-countable topological space is sequentially compact if and only if it is countably compact.
Proof
Follows directly from:
- Sequentially Compact Space is Countably Compact
- Countably Compact First-Countable Space is Sequentially Compact
$\blacksquare$
Axiom of Countable Choice
This theorem depends on the Axiom of Countable Choice, by way of Sequentially Compact Space is Countably Compact.
Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.
As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.
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: Countability Axioms and Separability