TITLE: Connection between connected and compact spaces?
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: There is a subspecialty of toplogy called continuum theory, which studies connected compact spaces.  See this for some general information.