Past exam of the functional Analysis and Banach and C*-Algebras course of the University of Oxford / 2008 / A / 3 / b / Solution

If the two ambient topologies are not comparable, neither is contained in the other. By the definition of inclusion of topologies, there is a ${\tau }_1$-open set absent from ${\tau }_2$, and a ${\tau }_2$-open set absent from ${\tau }_1$. These are the requested $U_1,U_2$, giving alternative (b). Together with the comparable case, this proves that at least one of the two alternatives always holds; it does not assume that the alternatives are mutually exclusive.