I'm reading Topology by Munkres, and I'm having some trouble with exercise 5 on page 92 (screenshot attached). 1. The problem statement, all variables and given/known data Let X and X' denote a single set in the topologies T and T', respectively; let Y and Y' denote a single set in the topologies U and U', respectively. Assume these sets are nonempty. Show that if T' is finer than T and U' is finer than U, then the product topology on X' x Y' is finer than the product topology on X x Y. 3. The attempt at a solution I think this is a counterexample... Let T = T' = U = U' be the standard topology on ℝ. Let X = Y = Y' = (0, 1). Let X' = (2, 3). X is open in X and Y is open in Y → X x Y is open in X x Y. X is not a subset of X' →X is not open in X' → X x Y is not open in X' x Y'. Hence the product topology on X' x Y' is not finer than the product topology on X x Y.