Products of Michael spaces and completely metrizable spaces

Abstract:

For disjoint subsets $A,C$ of $[0,1]$ the Michael space $M(A,C)=A\cup C$ 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 Lindelöf Michael space $M(A,C)$, of minimal weight $\aleph _1$, with $M(A,C)\times B(\aleph _0)$ Lindelöf but with $M(A,C)\times B(\aleph _1)$ not normal. ($B(\aleph _\alpha )$ denotes the countable product of a discrete space of cardinality $\aleph _\alpha$.) If $M(A)$ denotes $M(A,[0,1]\smallsetminus A)$, the normality of $M(A)\times B(\aleph _0)$ implies the normality of $M(A)\times S$ for any complete metric space $S$ (of arbitrary weight). However, the statement “$M(A,C)\times B(\aleph _1)$ normal implies $M(A,C)\times B(\aleph _2)$ normal” is axiom sensitive.