A cardinal invariant related to cleavability

A space X is κ-cleavable over Y if, for any partition of X into κ disjoint sets, there is a continuous function f:X→Y such that the images of these sets under f are pairwise disjoint. This notion defines a cardinal function on Y, namely the least κ such that whenever X is κ-cleavable over Y then there is a continuous injection X→Y. After a brief exploration of κ-cleavability in general, we investigate κ-cleavability over R2. We prove that a σ-compact polyhedron X is 6-cleavable over R2 if and only if X embeds in R2.