Covering versus partitioning with Polish spaces

TL;DR

This paper explores the relationship between covering and partitioning a completely metrizable space into Polish spaces, proving the consistency of strict inequalities under large cardinal assumptions and their necessity.

Contribution

It demonstrates the consistency of strict inequalities between covering and partitioning numbers under large cardinal assumptions and shows these assumptions are necessary.

Findings

Strict inequality $rak{cov}(X) < rak{par}(X)$ can be consistent with large cardinals.

Large cardinals are necessary for such strict inequalities to hold.

If the strict inequality holds, then $0^\

Abstract

Given a completely metrizable space $X$, let $\mathfrak{par}(X)$ denote the smallest possible size of a partition of $X$ into Polish spaces, and $\mathfrak{cov}(X)$ the smallest possible size of a covering of $X$ with Polish spaces. Observe that $\mathfrak{cov}(X) \leq \mathfrak{par}(X)$ for every $X$, because every partition of $X$ is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality $\mathfrak{cov}(X) < \mathfrak{par}(X)$ can hold for some completely metrizable space $X$. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if $\mathfrak{cov}(X) < \mathfrak{par}(X)$ for any completely metrizable $X$, then $0^\dagger$ exists.