Compact Hausdorff implies Bolzano

Question

Let $X$ be a compact Hausdorff topological space.

Is it true that every sequence has a converging subsequence?

Answer 1

No, there are compact Hausdorff spaces without any non-trivial (trivial meaning eventually constant) convergent subsequences,like $\beta \omega$, the Cech-Stone compactification of the natural numbers. compact non-sequentially compact spaces like $[0,1]^I$ for $I$ of size $|\mathbb{R}|$. For so-called sequential spaces, which include the first countable and the metric ones, this implication does hold. The Bolzano-Weierstrass peoperty is now called being sequentially compact, BTW.

Answer 2

"Every sequence has a convergent subsequence" is the property nowadays called "sequentially compact". It is not equivalent to "compact" for Hausdorff spaces. But it is equivalent for metric spaces.