Weak Reduction Principle and Computable Metric Spaces

This paper is a part of the ongoing research on developing a foundation for studying arithmetical and descriptive complexity of partial computable functions in the framework of computable topology. We propose new principles for computable metric spaces. We start with a weak version of Reduction Principle (\(\mathrm {WRP}\)) and prove that the lattice of the effectively open subsets of any computable metric space meets \(\mathrm {WRP}\). We illustrate the role of \(\mathrm {WRP}\) within partial computability. Then we investigate the existence of a principal computable numbering for the class of partial computable functions from an effectively enumerable space to a computable metric space. We show that while in general such numbering does not exist in the important case of a perfect computable Polish space it does. The existence of a principal computable numbering gives an opportunity to make reasoning about complexity of well-known problems in computable analysis in terms of arithmetical and analytic complexity of their index sets.