Dual discreteness of Σ-products and irreducibility of infinite products

Abstract

This paper includes two main results. Dual discreteness is a well known generalization of D-spaces. The first one is that every Σ-product of compact metric spaces is dually discrete. The property aD is another generalization of D-spaces, and it implies irreducibility. The second one is that the product $N^{{\omega }_1}$ of ${\omega }_1$ many copies of $N$ is irreducible, where $N$ denotes an infinite countable discrete space.