 Since was arbitrary, can be as large as desired by choosing appropriately.In the 1950s, Hillel Furstenberg introduced a proof by contradiction using point-set topology.Define a topology on the integers Z, called the evenly spaced integer topology, by declaring a subset "U" ⊆ Z to be an open set if and only if it is either the empty set, ∅, or it is a union of arithmetic sequences "S"("a", "b") (for "a" ≠ 0), where

Then a contradiction follows from the property that a finite set of integers cannot be open and the property that the basis sets "S"("a", "b") are both open and closed, since
cannot be closed because its complement is finite, but is closed since it is a finite union of closed sets.