It is a known fact that a 2nd countable compact Hausdorff space is metrizable. What if we weaken the 2nd countable to separable only - is the space still metrizable?

The core of the question, or a better formulation, is this: Is separability equivalent to 2nd countability in the class of compact Hausdorff spaces? (Similarly as it is in some classes of topological spaces and this certainly is not answered by Wikipedia as suggested below.)

The answer is NO. You can find examples of non-metrizable separable compact Hausdorff spaces below.

Counterexamples in Topology. π-Base is a more high-tech resource. $\endgroup$ – Nate Eldredge Aug 17 '18 at 18:16