Hi, the following is a well known theorem


*Let $M$ be a metric space. If every uncountable subset of $M$ has a limit point, then $M$ is separable.*


Question: Is there a similar result for topological spaces?


I have almost no knowledge of topology so I can only hope that this is not trivial.



Topological spaces, uncountable subsets and separability
The statement, "Let $M$ be a topological space. If every uncountable subset of $M$ has a limit point, then $M$ is separable," is false. Consider the first uncountable ordinal $\omega\_1$, under the order topology (see <http://en.wikipedia.org/wiki/First_uncountable_ordinal>). $\omega\_1$ is countably compact, hence also weakly countably compact (that is, every infinite subset has a limit point), but $\omega\_1$ is not separable.


