Covering versus partitioning with Polish spaces

Abstract

Given a completely metrizable space X, let par(X) denote the smallest possible size of a partition of X into Polish spaces, and cov(X) the smallest possible size of a covering of X with Polish spaces. Observe that cov(X) ≤ 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 cov(X) < 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 cov(X) < par(X) for any completely metrizable X, then 0 exists.