The breadth of constructibility degrees and definable Sierpi\'nski's coverings

TL;DR

This paper explores the relationship between the existence of certain definable coverings of Euclidean space and a specific cardinal invariant related to constructibility degrees, generalizing previous results in descriptive set theory.

Contribution

It extends prior work by Törnquist and Weiss, establishing a connection between $oldsymbol{ m oldsymbol{ ext{Sigma}}}_2^1$ Sierpiński coverings and the breadth invariant of constructibility degrees.

Findings

Established a link between definable coverings and the breadth invariant.

Generalized previous results to higher dimensions.

Provided new insights into the structure of constructibility degrees.

Abstract

Generalizing a result of T\"ornquist and Weiss, we study the connection between the existence of $\varSigma_2^1$ Sierpi\'{n}ski's coverings of $\mathbb{R}^n$, and a cardinal invariant of the upper semi-lattice of constructibility degrees known as breadth.