# 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.