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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 $\mathfrak {cov}(X) \lt \mathfrak {par}(X)$ can hold for some completely metrizable space $X$. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if $\mathfrak {cov}(X) \lt \mathfrak {par}(X)$ for any completely metrizable $X$, then $0^\dagger $ exists.