Minimal bounds and members of effectively closed sets

TL;DR

This paper investigates the properties of effectively closed sets, demonstrating the existence of a special class where no member acts as a minimal cover, thus impacting the understanding of degrees of minimal covers in computability theory.

Contribution

It proves that degrees of minimal covers do not form a basis for non-empty special a^0_1 classes, revealing new structural insights.

Findings

Existence of a non-empty special a^0_1 class with no minimal cover members

Degrees of minimal covers cannot serve as a basis for a^0_1 classes

Clarifies the relationship between minimal covers and effectively closed sets

Abstract

We show that there exists a non-empty special $\Pi^0_1$ class in which no member is a minimal cover for any set, hence prove that degrees of minimal covers cannot be a basis for $\Pi^0_1$ classes.