# Topological spaces, uncountable subsets and separability

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.

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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.

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Consider the topological space $[0, \omega_1]$ where $\omega_1$ is the first uncountable ordinal. Every uncountable subset has a limit point, namely $\omega_1$, because the complement of any neighbourhood of $\omega_1$ is countable. However, it is not separable, since the supremum of a countable subset of countable ordinals is countable.

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Every compact space satisfies "every infinite set has a limit point". (I am assuming here that by limit point you mean what I would call an accumulation point, i.e., a point $x$ such that evey neighborhood contains infinitely many elements of the set.) So in particular, in a compact space every uncountable set has a limit point. It follows that every compact space that is not separable (for instance a sufficiently high power of the closed unit interval or Robert Israel's example) shows that the theorem you mention does not hold for topological spaces in general.