Let $(X,d)$ be a complete metric space. If $X$ has no isolated points then $X$ is uncountable.
I know that for a Hausdorff and compact general space it is true but, how do I use Baire category in this specific case?
Thanks a lot.
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1This answer contains a proof of the stronger result that $|X|$ is at least $2^\omega=\mathfrak{c}$. – Brian M. Scott Dec 04 '15 at 16:32
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nice @BrianM.Scott!!! Thanks. – L.F. Cavenaghi Dec 04 '15 at 16:34
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My pleasure! $,$ – Brian M. Scott Dec 04 '15 at 16:34
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@BrianM.Scott, just one questions, is him assuming Hausdorff on this proof, right? To conclude that this intersection is empty? – L.F. Cavenaghi Dec 04 '15 at 16:38
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It’s a metric space, so it’s automatically Hausdorff. – Brian M. Scott Dec 04 '15 at 16:40
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Yes, indeed. Thanks again. – L.F. Cavenaghi Dec 04 '15 at 16:41