A blend of methods of recursion theory and topology: A Π10 tree of shadow points

Abstract

This paper is a sequel to our [7]. In that paper we constructed a Π10 tree of avoidable points. Here we construct a Π10 tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2<ω. One interesting fact is that while our Π10 tree of sharp filters (a subtree of Δ<ω) is isomorphic to the tree of addresses (a subtree of 2<ω), the tree of addresses is recursively enumerable but not recursive. To achieve this end we use a finite injury priority argument.