Connections between Kuratowski partitions of Baire spaces, measurable cardinals and precipitous ideals

TL;DR

This paper explores the properties of $K$-partitions in Baire spaces and their links to measurable cardinals and precipitous ideals, revealing implications for topology and set theory.

Contribution

It establishes the existence of $K$-partitions in various spaces and connects these to large cardinal hypotheses and ideal properties, advancing understanding of their interplay.

Findings

Existence of $K$-partitions implies similar partitions in metrizable and compact spaces.

Connections established between $K$-partitions, precipitous ideals, and measurable cardinals.

Potential links with real-measurable cardinals and measure extensions are discussed.

Abstract

In this paper we present a few properties of $K$-partitions, which are partitions of Baire spaces such that all subfamilies of such a partition sum to a set with the Baire property. Among the result proven we have general existence result that state that the existence of any $K$-partition implies the existence of $K$-partition of a metrizable space as well as existence of $K$-partition of a compact space implies the existence of $K$-partition of a completely metrizable space. We also prove some connections between existence of $K$-partitions and existence of precipitous ideals as well as measurable cardinals. There are also outlined possible connection with real-measurable cardinals, extensions of Lebesgue measure on the closed interval and density topologies.