Abstract topological dynamics involving set-valued functions

Continuous functions over compact Hausdorff spaces have been completely characterised. We consider the more general problem: given a set-valued function T from an arbitrary set X to itself, does there exist a compact Hausdorff topology on X with respect to which T is upper semicontinuous? We give conditions that are necessary for T to be upper semicontinuous and point-closed if X is a compact Hausdorff space. We show that it is always possible to provide X with a compact T₁ topology with respect to which T is lower semicontinuous, and consequently, if T:X - X is a function, then it is always possible to provide X with a compact T₁ topology with respect to which T is continuous.