The ultrafilter theorem, UFT, is the simple statement that every filter can be extended to an ultrafilter. This statement is independent of ZF set theory, is implied by the axiom of choice, yet is strictly weaker than the axiom of choice. It turns out to be equivalent over ZF to many compactness theorems in various subfields of mathematics. We'll show that UFT is equivalent to:

1) In topology, Tychonoff's theorem for Hausdorff spaces.

2) In mathematical logic, the compactness theorem for propositional logic.

3) In functional analysis, the weak-star compactness of the unit ball of a dual space.