Fréchetness of inverse limits and products
A class \({\mathcal C}\) of spaces is almost countably productive provided that for every sequence \(\{X_ n\}\) of spaces, the product of all the \(X_ n\) is in \({\mathcal C}\) whenever each finite product of the \(X_ n\) is in \({\mathcal C}\). Furthermore \({\mathcal C}\) is countably projective provided that the inverse limit is in \({\mathcal C}\) for each inverse sequence from \({\mathcal C}\) with surjective bonding maps. The latter implies the former. This paper shows that each of the six classes of spaces \(<\alpha_ 1>\), \(<\alpha_ 2>\), \(<\alpha_ 3>\), \(<\alpha_ 1-FU>\), \(<\alpha_ 2-FU>\) and \(<\alpha_ 3-FU>\) [due to \textit{A. V. Arkhangel'skij}, Dokl. Akad. Nauk SSSR 206, 265-268 (1972; Zbl 0275.54004)] is countably projective. Here \(<\alpha_ i-FU>\) stands for the intersection of \(<\alpha_ i>\) and the class of Fréchet-Urysohn spaces. The class \(<\alpha_ 2>\) is precisely the class of w-spaces. So this answers a question of Galvin as to whether the class of w-spaces is countably productive.
scientific article; zbMATH DE number 3977907
scientific article