Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo–Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums ( = coproducts ) and products. However, it may happen that for families ( X i ) i ∈ I and ( Y i ) i ∈ I with the property that each X i is homeomorphic to the corresponding Y i neither their sums ⊕ i ∈ I X i and ⊕ i ∈ I Y i nor their products ∏ i ∈ I X i and ∏ i ∈ I Y i are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.