Sequentially Compact Space is Countably Compact

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Theorem

A sequentially compact topological space is also countably compact.


Proof

Let $T = \left({S, \tau}\right)$ be a sequentially compact topological space.

By definition of countably compact space, it suffices to show that every infinite sequence in $S$ has an accumulation point in $S$.


By the definition of a sequentially compact space, every infinite sequence in $S$ has a subsequence which converges in $S$.

The result follows from Limit Point of Sequence is Accumulation Point.

$\blacksquare$


Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice, by way of Equivalence of Definitions of Countably Compact Space.

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