  CW complexes are used extensively in algebraic topology, but the product of
two CW complexes need not be a CW complex, as shown by Dowker. Whilst Whitehead
and Milnor gave sufficient conditions for the product to be a CW complex, all
existing characterizations of those pairs of CW complexes which have product a
CW complex rely on extra set-theoretic axioms. Here we provide a complete
characterization of when the product of two CW complexes is a CW complex
without using any extra set-theoretic axioms.
