A question in general topology

Recall the following theorem: Given a set {X} with two topologies {\cal U} and {\cal V}, with {\cal U} weaker than {\cal V}, if {\cal U} is Hausdorff and {\cal V} is compact then in fact {\cal U = \cal V}.

In general, is there a “right” topology, in this sense? Specifically, given a Hausdorff topology {\cal U} can you weaken it to a compact topology which is still Hausdorff, and given a compact topology {\cal V} can you strengthen it to a Hausdorff topology which is still compact?

For example, consider the cofinite topology on {\omega}. The cofinite topology is always compact. Can we extend this topology to a Hausdorff topology? Yes we can: {\omega} could just as well have been any countably infinite set, say {\{0\}\cup\{1/n: n\in{\bf Z}^+\}}, and now one checks that the usual topology (induced by {\mathbf{R}}, 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.

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