FALSE: every sequentially compact space is compact

False claim: every sequentially compact topological space (Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets) is compact (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Where the claim comes from, and what is actually true. For a metric space the two conditions are equivalent, and that equivalence is proved elsewhere in this library at a stated choice cost; the claim above is that equivalence transplanted to an arbitrary topological space, where it fails. What does hold in general is only that sequential compactness implies countable compactness, and that at the cost of countable choice.

The refutation assumes the Axiom of Countable Choice (The Axiom of Countable Choice ($\mathrm{AC}_\omega$)), because that is what makes the witness sequentially compact; without it the witness is not known to have the property the claim would have to preserve. The witness is $\omega_1$ with its order topology (The first uncountable ordinal $\omega_1 := \aleph(\omega)$, On an ordinal with its order topology the sets $[0,\beta]$ and $(\alpha,\beta]$ form a basis of clopen sets, the isolated points are exactly the non-limit ordinals, and the space is Hausdorff).

Why the two conditions can diverge at all. Sequential compactness tests countably many points at a time and compactness tests covers of any size. In $\omega_1$ a sequence is a countable object and is therefore bounded below $\omega_1$, while the cover by initial segments is uncountable and climbs the whole ordinal; the two conditions are simply looking at different cardinalities. For a metric space the topology is determined by countably many balls at each point and the divergence disappears.

The implication that does survive is sequential compactness to countable compactness, assuming countable choice (Compact implies countably compact, Lindel"of and limit point compact; countably compact together with Lindel"of implies compact; and, at the cost of countable or dependent choice, sequentially compact implies countably compact, countably compact implies limit point compact, and the converse holds when every singleton is closed, claim 2).

The converse claim also fails, and its witness is a different space entirely: a compact space that is not sequentially compact is exhibited in FALSE: every compact space is sequentially compact. Neither of the two implications holds in general, so sequential compactness and compactness are incomparable conditions on topological spaces.