Continuous maps of products of metrizable spaces (Q2716443)

Generalizing \textit{V. Trnková}'s solution [Trans. Am. Math. Soc. 343, No. 1, 305-325 (1994; Zbl 0803.54015)] of W. Taylor's problem for clones, the authors show that for any nonempty set \(S\) and any convenient subset \(Z\) of the direct sum \(E(S)=\sum_S(N,+)\) one can find families \(\{X_s\}_S, \{Y_s\}_S\) of metrizable spaces such that for any convenient \(A\subset E(S)\) the full subcategories of TOP consisting of finite products \(\prod_S \{X_s^{\varphi(s)}\}\), \(\varphi\in A\) or \(\prod_S \{Y_s^{\varphi(s)}\}\), \(\varphi\in A\), resp., are isomorphic iff \(A\subset Z\).