This paper studies the problem whether there is a nontrivial productive coreflective class of topological spaces (i.e., a class closed under all products, sums and quotients). A subclass \({\mathcal C}\) of a category \({\mathcal K}\) is \(\kappa\)-productive if every product (in \({\mathcal K}\)) of less than \(\kappa\) objects of \({\mathcal C}\) belongs to \({\mathcal C}\), and \({\mathcal C}\) is finitely (countably) productive if it is \(\omega- (\omega_1\)-) productive. The productivity number of a subclass \({\mathcal C}\) of \({\mathcal K}\) is the smallest cardinal \(\kappa\) (if it exists) such that a product in \({\mathcal K}\) of \(\kappa\) many objects of \({\mathcal C}\) does not belong to \({\mathcal C}\); if no such cardinal exists, then \({\mathcal C}\) is productive. Let Top be the category of topological spaces and continuous maps. The author proves that a finitely productive coreflective subclass \({\mathcal C}\) of Top is \(\kappa\)-productive, where \(\kappa\geq\omega\), if and only if the generalized Cantor set \(2^\lambda\) is in \({\mathcal C}\) for all \(\lambda<\kappa\). An infinite cardinal \(\kappa\) is submeasurable if there exists a noncontinuous, \(\kappa\)-continuous real-valued map on \(2^\kappa\), where a map is \(\kappa\)-continuous if it preserves limits of well-ordered nets of length less than \(\kappa\). It is also proved that every submeasurable cardinal is a productivity number of some coreflective class of Top, and that productivity numbers of finitely productive and nonproductive coreflective classes in Top are submeasurable cardinals. Let \({\mathfrak s}\) denote the first sequential cardinal. The last result implies that if a coreflective class in Top is countably productive, then it is \({\mathfrak s}\)-productive, thus, it is productive if \({\mathfrak s}\) does not exist in the model we work in.