CW spaces are often presented as the "spaces of choice" in algebraic topology courses, being relatively nice spaces built up by successively gluing on Euclidean balls of increasing dimension. However, the product of CW complexes need not be a CW complex, as shown by Dowker soon after CW complexes were introduced. Work in the 1980s characterised when the product is a CW complex under the assumption of CH, or just \(b = \aleph_1\). In this talk I will give and prove a complete characterisation of when the product of CW complexes is a CW complex, valid under ZFC. The characterisation however involves \(b\); the proof is point-set-topological (I won't assume any knowledge of algebraic topology) and uses Hechler conditions.
