# Connection between connected and compact spaces?

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.