= Smash product of CW pairs
{title2=$(X,A)\wedge(Y,B)=(X\times Y,(A\times Y)\cup(X\times B))$}
For <CW pairs> $(X,A)$ and $(Y,B)$, write their product pair as
$$
\bigl(X\times Y,\ (A\times Y)\cup(X\times B)\bigr).
$$
Use the \b[<compactly generated product>], equivalently the weak <CW complex> topology on products of cells. For arbitrary infinite <CW complexes> this can be finer than the ordinary <product topology>. The ordinary <product topology> already agrees when one factor is locally compact, or when both factors have countably many cells; in particular there is no distinction for <finite CW complexes>.<Expanded Appendix to Algebraic Topology by Allen Hatcher>[Theorem A.6, pages 6–7]{ref}
The quotient of this pair is $(X/A)\wedge(Y/B)$. Setwise, the collapsed subset is exactly the <set> of pairs with at least one coordinate in its collapsed subspace; the compatible quotient constructions give the <homeomorphism> in the <compactly generated space> convention. In a multiplicative <generalized cohomology theory>, external products of relative classes of degrees $k,m$ lie in degree $k+m$ for this pair. Pullback along a <diagonal map> gives the <relative cup product>.
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