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What kind of connection is there between connected and compact subspaces, if any?

I am just curious. I know that the image of a compact space under a continuous function is compact and the same holds for connected spaces. But this is not what I am looking for. I would like to see a condition on a set or a topological space in order to see if we can infer that if a set is connected + some other condition then it is compact. (Something similar to any compact set in a Hausdorff space is closed. In this case the extra condition would be having a Hausdorff space).

(Sorry for the initial confusion. I meant compact set, not connected, in my example.) Thanks.

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There are no such connections.

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  • $\begingroup$ A fairly bold claim! =] $\endgroup$ – Tara B Feb 19 '13 at 23:09
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    $\begingroup$ @TaraB: And I y'aint afraid to make it! $\endgroup$ – gnometorule Feb 19 '13 at 23:40
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There is a subspecialty of toplogy called continuum theory, which studies connected compact spaces. See this for some general information.

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