A cardinal invariant related to cleavability (Q1935856)

Given \(X\) and \(Y\) topological spaces, we say that \(X\) is \textit{cleavable} over \(Y\) if for any subset \(A \subseteq X\) there is a continuous function \(f: X \rightarrow Y\) such that the images of the disjoint sets \(A\) and \(X \setminus A\) are also disjoint [\textit{A. V. Arhangel'skiĭ}, Topology Appl. 44, No. 1--3, 27-36 (1992; Zbl 0786.54014)]. The author generalizes this notion for any cardinal \(\kappa\) by saying that \(X\) is \(\kappa\)-cleavable over \(Y\) if for every partition of \(X\) into \(\kappa\) disjoint sets there is a continuous function \(f:X \rightarrow Y\) such that the images of these sets under \(f\) are pairwise disjoint; he also introduces a cardinal invariant \(\clubsuit(X,Y)\), defined as the least cardinal such that \(X\) is not \(\kappa\)-cleavable over \(Y\). It is also shown in the paper that, for any space \(Y\) and for any class \(\mathcal{C}\) of topological spaces, there is a cardinal \(\kappa\) such that whenever \(X \in \mathcal{C}\) is \(\kappa\)-cleavable over \(Y\), then \(X\) \textit{condenses} into \(Y\) (meaning that there is a continuous injection from \(X\) into \(Y\)). The least cardinal satisfying the preceding phrase is defined to be the \textit{cleavage} of \(Y\) relative to the class \(\mathcal{C}\) and is denoted by \(\clubsuit_{\mathcal{C}}(Y)\). The main results of the paper are: {Theorem 2.} For every cardinal \(\mu \geqslant 2\) there is a pair \((X,Y)\) of Hausdorff spaces such that \(\clubsuit(X,Y) = \mu\). {Theorem 12.} Any \(\sigma\)-compact polyhedron which is \(6\)-cleavable over \(\mathbb{R}^2\) embeds in \(\mathbb{R}^2\). The preceding theorem translates as \(\clubsuit_{\sigma CP}(\mathbb{R}^2) = 6\), where \(\sigma CP\) is the class of \(\sigma\)-compact polyhedra. {Theorem 21.} \(\clubsuit(D_{3,3}) = 3\), where \(D_{3,3}\) is the topological space obtained by expanding each vertex of the graph \(K_{3,3}\) to a copy of the closed unit disc. The paper ends by posing some questions, for instance: is there a compact space \(X\) which does not embed in \(\mathbb{R}^2\) but satisfies \(\clubsuit(X,\mathbb{R}^2) > 6\) ?