Covering versus partitioning with the Cantor space (Q2105041)

In this extremely pleasant paper, the author investigates the relationship between ``being coverable'' and ``being partitionable'', for various topological spaces. If \(X\) and \(Y\) are topological spaces, we say that \(X\) is \textit{\(Y\)-coverable} if \(X\) can be covered by homeomorphic copies of \(Y\); and we say that \(X\) is \textit{\(Y\)-partitionable} if it can be partitioned into homeomorphic copies of \(Y\). Thus, clearly, being \(Y\)-partitionable implies being \(Y\)-coverable, and the paper under review explores various circumstances under which the converse implication holds. After the first section containing an introduction, Section 2 establishes as its main result that every first countable space \(X\) with \(|X|\leq\mathfrak c\) is \(2^\omega\)-partitionable iff it is \(2^\omega\)-coverable. There is also a nice example showing that the condition \(|X|\leq\mathfrak c\) is necessary, and the bound of \(\mathfrak c\) is sharp. The next section's main result is that, for a metrizable space \(X\) satisfying \(|X|\leq\mathfrak c^{+\omega}\), being \(2^\omega\)-partitionable is equivalent to being \(2^\omega\)-coverable. The section also contains several lemmas leading to the construction of an example showing that the condition \(|X|\leq\mathfrak c^{+\omega}\) is necessary, and the bound of \(\mathfrak c^{+\omega}\) is sharp. Finally, in Section 4, the main result states that, for a completely metrizable space \(X\), four conditions are equivalent: being \(2^\omega\)-partitionable, being \(2^\omega\)-coverable, having no nonempty open \(\sigma\)-discrete subspace, and having no isolated points. There is, at the end, a short final section showing that the previous results (from all previous sections) remain true when one replaces \(2^\omega\) with any zero-dimensional Polish space without isolated points. A few final comments establish that the hypothesis that the incumbent space contains no isolated points is necessary, even though, on the other hand, for every zero-dimensional Polish space \(Y\) (regardless of whether it has isolated points), every \(2^\omega\)-partitionable space must be \(Y\)-partitionable as well.