Abstract

A Cantor singleton is the unique nonrecursive member of some $\Pi^0_1$ class. In this paper we investigate the relationships between the following three notions: Cantor singletons, Cantor-Bendixson rank, and recursive join. Among other results, we show that the rank of $A\oplus B$ is at most the natural sum of the ranks of $A$ and $B$ , and that, if $B$ has the same rank as $A\o plus B$ , then $A$ is recursive in $B$