Covering versus partitioning with the Cantor space

Abstract

What topological spaces can be partitioned into copies of the Cantor space 2ω? An obvious necessary condition is that a space can be partitioned into copies of 2ω only if it can be covered with copies of 2ω. We prove three theorems concerning when this necessary condition is also sufficient.If X is a metrizable space and |X|≤c+ω (the least limit cardinal above c), then X can be partitioned into copies of 2ω if and only if X can be covered with copies of 2ω. To show this cardinality bound is sharp, we construct a metrizable space of size c+(ω+1) that can be covered with copies of 2ω, but not partitioned into copies of 2ω.Similarly, if X is first countable and |X|≤c, then X can be partitioned into copies of 2ω if and only if X can be covered with copies of 2ω. On the other hand, there is a first countable space of size c+ that can be covered with copies of 2ω, but not partitioned into copies of 2ω.Finally, we show that, regardless of its cardinality, a completely metrizable space can be partitioned into copies of 2ω if and only if it can be covered with copies of 2ω if and only if it has no isolated points.