Given a set does there exist a topology on that set that makes it compact?

Question

Say you are given a set X, does there always exist a non-trivial (not the coarse topology) topology on X such that X is

1) Comapact 2) Connected 3) Compact and Connected

Additionally, if it were possible can you give an example of such a topology on the set of Integers(where in the set of integers would be compact,connected)