Recall the following theorem: Given a set with two topologies and , with weaker than , if is Hausdorff and is compact then in fact .
In general, is there a “right” topology, in this sense? Specifically, given a Hausdorff topology can you weaken it to a compact topology which is still Hausdorff, and given a compact topology can you strengthen it to a Hausdorff topology which is still compact?
For example, consider the cofinite topology on . The cofinite topology is always compact. Can we extend this topology to a Hausdorff topology? Yes we can: could just as well have been any countably infinite set, say , and now one checks that the usual topology (induced by , say) is compact, Hausdorff, and stronger than the cofinite topology (because finite sets are closed).
EDIT: A more or less complete answer can be found in this MO answer.