Connection between connected and compact spaces?

Question

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.

score 8 · Accepted Answer · answered Feb 19 '13 at 22:55

8

There are no such connections.

answered Feb 19 '13 at 22:55

gnometorule

4,640

1

A fairly bold claim! =] – Tara B Feb 19 '13 at 23:09
1

@TaraB: And I y'aint afraid to make it! – gnometorule Feb 19 '13 at 23:40

score 4 · Answer 2 · answered Feb 19 '13 at 23:10

4

There is a subspecialty of toplogy called continuum theory, which studies connected compact spaces. See this for some general information.

answered Feb 19 '13 at 23:10

ncmathsadist

49,383

Connection between connected and compact spaces?

2 Answers2