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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.

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closed as off-topic by Emil Jeřábek, Will Brian, David Handelman, Jan-Christoph Schlage-Puchta, Mateusz Kwaśnicki Aug 19 '18 at 23:09

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  • $\begingroup$ I apologize for the question and wasting your time - I should have easily found the answer myself. $\endgroup$ – Roman Hric Aug 17 '18 at 17:29
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    $\begingroup$ It was fun to several of us. $\endgroup$ – Wlod AA Aug 17 '18 at 17:38
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    $\begingroup$ Standard place to look is the book Counterexamples in Topology. π-Base is a more high-tech resource. $\endgroup$ – Nate Eldredge Aug 17 '18 at 18:16
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    $\begingroup$ Next ask whether a first-countable compact Hausdorff space is metrizable. $\endgroup$ – Gerald Edgar Aug 17 '18 at 21:32
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The Cech-Stone compactification of the integers is by the very definition separable (as it contains the countable dense set of integers) yet it is hardly metrisable as it contains a non-separable subspace (remove the integers).

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From Wikipedia: a compact Hausdorff space is metrizable if and only if it is second-countable. See the proof on PlanetMath.

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    $\begingroup$ See also math.stackexchange.com/q/83652, math.stackexchange.com/q/1642751, math.stackexchange.com/q/573805, math.stackexchange.com/q/234018. Even more counterexamples here: math.stackexchange.com/q/74923. A web search (like I did) would have made posting your question needless. $\endgroup$ – Alex M. Aug 17 '18 at 17:19
  • $\begingroup$ I should have formulated my question as follows: Is separability equivalent to 2nd countability in compact Hausdorff spaces? Sure I did the web search but didn't succeed to find the counterexamples. $\endgroup$ – Roman Hric Aug 17 '18 at 18:37
  • $\begingroup$ Since second-countability is equivalent to metrizability, and since there are separable compact Hausdorff spaces that are not metrizable (see my couterexamples link), it follows that separability is weaker than second-countability. $\endgroup$ – Alex M. Aug 17 '18 at 18:42
  • $\begingroup$ Of course, it was clear after opening your link to counterexamples. Anyway, even the question might be dumb, the answers might be useful also to someone else. $\endgroup$ – Roman Hric Aug 17 '18 at 18:49
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Alexandrov double arrow space is a standard example of a compact Hausdorff separable space that is not metrizable.

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I can't resist.

Urysohn (followed by Bing and others) has constructed a connected Hausdorff space which has only countably many points but more than one. Of course, such spaces are separable.

On the other hand, every connected component of an arbitrary countable metric space has only one point.

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