On Turing degrees of points in computable topology

Abstract This paper continues our study of computable point‐free topological spaces and the metamathematical points in them. For us, a point is the intersection of a sequence of basic open sets with compact and nested closures. We call such a sequence a sharp filter . A function f F from points to points is generated by a function F from basic open sets to basic open sets such that sharp filters map to sharp filters. We restrict our study to functions that have at least all computable points in their domains. We follow Turing's approach in stating that a point is computable if it is the limit of a computable sharp filter; we then define the Turing degree Deg( x ) of a general point x in an analogous way. Because of the vagaries of the definition, a result of J. Miller applies and we note that not all points in all our spaces have Turing degrees; but we also show a certain class of points do. We further show that in ℝ n all points have Turing degrees and that these degrees are the same as the classical Turing degrees of points defined by other researchers. We also prove the following: For a point x that has a Turing degree and lies either on a computable tree T or in the domain of a computable function f F , there is a sharp filter on T or in dom( F ) converging to x and with the same Turing degree as x . Furthermore, all possible Turing degrees occur among the degrees of such points for a given computable function f F or a complete, computable, binary tree T . For each x ∈ dom( f F ) for which x and f F ( x ) have Turing degrees, Deg( f F ( x )) ≤ Deg( x ). Finally, the Turing degrees of the sharp filters convergent to a given x are closed upward in the partial order of all Turing degrees. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)