Paracompact product spaces defined by ultrafilters over the index set

Abstract:

Let $\omega = \{ 0,1, \ldots \}$, and suppose that for each $i \in \omega ,{C_i}$ is a compact Hausdorff space with weight $\leq c$. A filter over $\omega$ defines a topology on $\prod \nolimits _{i \in \omega } {{C_i}}$. We prove that the continuum hypothesis implies the existence of ultrafilters over $\omega$ for which the corresponding product space on $\prod \nolimits _{i \in \omega } {{C_i}}$ is paracompact. Moreover, we show that every ${\mathbf {P}}$-point in $\beta \omega - \omega$ is an ultrafilter with this property. Since box products appear as closed subspaces of ultrafilter products, our theorem extends results of Mary Ellen Rudin (1972) and Kenneth Kunen (1978).