The product of 〈αi〉-spaces

AbstractThe purpose of this paper is to give answers to the following problems posed by A.V. Arhangel'skii: Is the product of 〈αi〉-spaces 〈αi〉 for i = 1, 2, 3, 4, 5? Is every countably compact sequential space 〈αi〉? We also give, under CH, a negative answer to the following problem: If a subspace of the product of finitely many strongly Fréchet spaces is Las̆nev, then is it metrizable