使用波兰空间进行覆盖与分区

IF 0.5 3区数学 Q3 MATHEMATICS

Fundamenta Mathematicae Pub Date : 2021-01-25 DOI:10.4064/fm28-5-2022

W. Brian

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引用次数: 6

摘要

给定一个完全可度量的空间X，设par（X）表示X到波兰空间的分区的最小可能大小，cov（X）是X与波兰空间的覆盖的最小可能尺寸。观察每个X的cov（X）≤par（X），因为X的每个分区也是一个覆盖。我们证明了严格不等式cov（X）本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Covering versus partitioning with Polish spaces

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.