Comparing the degrees of enumerability and the closed Medvedev degrees

TL;DR

This paper explores the relationships between degrees of enumerability and closed Medvedev degrees, revealing complex interactions and characterizations, including the correspondence between compact degrees of enumerability and cototal enumeration degrees.

Contribution

It provides a detailed comparison of enumerability and closed Medvedev degrees, identifying their non-inclusion relations and characterizing compact degrees of enumerability.

Findings

Nonzero closed degrees do not always bound nonzero degrees of enumerability.

Nonzero degrees of enumerability do not always bound nonzero closed degrees.

Compact degrees of enumerability are exactly the cototal enumeration degrees.

Abstract

We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.