Covering versus partitioning with the Cantor space

Abstract

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