## Characterizing the continuous degrees

### Abstract

The continuous degrees measure the computability-theoretic content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees. Although nontotal (i.e., non-Turing) continuous degrees exist, they are all very close to total: joining a continuous degree with a total degree that is not below it always results in a total degree. We call this curious property almost totality.

We prove that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of enumeration degrees (Cai, Ganchev, Lempp, Miller, Soskova), we see that the continuous degrees are also definable. Applying earlier work on the continuous degrees, this shows that the relation PA above'' on the total degrees is definable in the enumeration degrees.

In order to prove that every almost total degree is continuous, we pass through another characterization of the continuous degrees that slightly simplifies one of Kihara and Pauly. Like them, we identify our almost total degree as the degree of a point in a computably regular space with a computable dense sequence, and then we apply the effective version of Urysohn's metrization theorem (Schröder) to reveal our space as a computable metric space.

This is joint work with Uri Andrews, Greg Igusa, and Mariya Soskova.