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unfortunately I am completely stuck on the follwing question:

Given a non compact metric space M that contains a non empty open compact subset, then M is not connected.

What examples are there?

How can I prove it?

Thank you very much in advance!

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  • $\begingroup$ what would be an example? I am trying to find something in the real numbers, but I do not manage to construct a set that is both open and compact. $\endgroup$ – Hans Apr 14 '14 at 17:57
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Hint: In the case where $M$ is metric, every compact set is closed. $M$ is connected iff every open and closed subset is either $M$ or $\emptyset$.

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    $\begingroup$ Every compacet set in a metric space (or more generally a Hausdorff space) is closed in it. You certainly implied this, but I think it’s worth stating it explicitly. $\endgroup$ – k.stm Apr 14 '14 at 16:05
  • $\begingroup$ yes, thanks for your comment. $\endgroup$ – mookid Apr 14 '14 at 16:08

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