On the Baire space of ${\omega }_1$-strongly compact weight

Abstract

We prove that on the Baire space $(D^{\kappa },\pi )$, $\kappa \ge {\omega }_0$ where $D$ is a uniformly discrete space having ${\omega }_1$-strongly compact cardinal and $\pi$ denotes the product uniformity on $D^{\kappa }$, there exists a $z_u$-filter $\mathcal{F}$ being Cauchy for the uniformity $e\pi$ having as a base all the countable uniform partitions of $(D^{\kappa },\pi )$, and failing the countable intersection property. This fact is equivalent to the existence of a non-vanishing real-valued uniformly continuous function $f$ on $D^{\kappa }$ for which the inverse function $g=1/f$ cannot be continuously extended to the completion of $(D^{{\kappa }_0},e\pi )$. This does not happen when the cardinal of $D$ is strictly smaller than the first Ulam-measurable cardinal.