Products of $\omega ^*$ images

Abstract:

Let $\omega ^*$ be the C̆ech-Stone remainder $\beta \omega \setminus \omega$. We show that there exists a large class $\mathcal {O}$ of images of $\omega ^*$ such that whenever $\mathcal {S}$ is a subset of $\mathcal {O}$ of cardinality at most the continuum, then $\omega ^* \times \prod \mathcal {S}$ is again an image of $\omega ^*$. The class $\mathcal {O}$ contains all separable compact spaces, all compact spaces of weight at most $\omega _1$ and all perfectly normal compact spaces.