Rank, join, and Cantor singletons (Q1387096)

In this paper, the relationship between the notions of Cantor singletons, Cantor-Bendixson rank and recursive join are investigated. In the first part of this paper, the author proves: If \(A\) is a Cantor singleton and \(B\) is a nonrecursive set and \(B\leq_{\text{tt}}A\), then \(B\) is a Cantor singleton. If \(A\oplus B\) is a Cantor singleton, then either \(B\) is recursive or \(A\) is recursive in \(B\). Let \(| A|\) be the Cantor-Bendixson rank of \(A\). In the second part, the relationship between \(| A\oplus B|\) and the ordinals \(| A|\), \(| B|\) is investigated.