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

TL;DR

This paper explores the properties of the Baire space over a set with an $oldsymbol{ ext{ω}_1}$-strongly compact cardinal, revealing the existence of special filters and functions related to uniform continuity and completions.

Contribution

It demonstrates the existence of a specific $z_u$-filter and a non-extendable inverse function on the Baire space with an $oldsymbol{ ext{ω}_1}$-strongly compact cardinal, highlighting a cardinal-dependent phenomenon.

Findings

Existence of a $z_u$-filter failing the countable intersection property.

Existence of a non-extendable inverse function for certain uniformly continuous functions.

Such phenomena do not occur below the first Ulam-measurable cardinal.

Abstract

We prove that on the Baire space $(D^{\kappa},\pi)$, $\kappa \geq \omega_0$ 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.