Products of Michael spaces and completely metrizable spaces

Abstract

For disjoint subsets A,C of [0, 1] the Michael space M(A,C) = A ∪ C has the topology obtained by isolating the points in C and letting the points in A retain the neighborhoods inherited from [0, 1]. We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelof Michael space M(A,C), of minimal weight א1, with M(A,C) × B(א0) Lindelof but with M(A,C) × B(א1) not normal. (B(אα) denotes the countable product of a discrete space of cardinality אα.) If M(A) denotes M(A, [0, 1]rA), the normality of M(A)×B(א0) implies the normality of M(A)× S for any complete metric space S (of arbitrary weight). However, the statement “M(A,C)×B(א1) normal implies M(A,C)×B(א2) normal” is axiom sensitive.