Is a compact Hausdoff space peaty?

This kind of topological space has a property: all limits of a sequence (or a net) that should exist, does exist, and they exist uniquely.
Now let Bool be a category of boolean algebra, One can define an ultrafilter monad from the composition of the hom functor ( hom(,2): Bool^op > set ) and its left adjoint.
This gives rise to an important question in Ray peat’s way of diets, Is such space peaty given that we can topologize the ultrafilter monad above by equipping it with a set, Would the set have to be open in order for the metabolism process to function correctly?