On resolvability of products

Abstract

. All spaces below are T 0 and crowded (i.e. have no isolated points). For n ≤ ω let M ( n ) be the statement that there are n measurable cardinals and Π( n ) ( Π + ( n ) ) that there are n +1 (0-dimensional T 2 ) spaces whose product is irresolvable. We prove that M (1) , Π(1) and Π + (1) are equiconsistent. For 1 < n < ω we show that CON ( M ( n )) implies CON (Π + ( n )) . Finally, CON ( M ( ω )) implies the consistency of having infinitely many crowded 0-dimensional T 2 -spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin from [11]. Concerning an even older question of Ceder and Pearson in [1], we show that the following are consistent modulo a measurable cardinal: These significantly improve Eckertson