On degree-preserving homeomorphisms between trees in computable topology

Abstract

In this paper we first give a variant of a theorem of Jockusch–Lewis– Remmel on existence of a computable, degree-preserving homeomorphism between a bounded strong \({\Pi^0_2}\) class and a bounded \({\Pi^0_1}\) class in 2ω. Namely, we show that for mathematically common and interesting topological spaces, such as computably presented \({\mathbb{R}^n}\) , we can obtain a similar result where the homeomorphism is in fact the identity mapping. Second, we apply this finding to give a new, priority-free proof of existence of a tree of shadow points computable in 0′.