On degree-preserving homeomorphisms between trees in computable topology

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 2 ^ω . 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′.