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The breadth of constructibility degrees and definable Sierpiński's coverings

Alessandro Andretta, Lorenzo Notaro

TL;DR

The paper deepens the link between geometric coverings and set-theoretic structure by showing that the existence of definable Sierpiński coverings is tightly controlled by the breadth of the upper semi-lattice of constructibility degrees. It introduces a real-breadth dichotomy, proves a definable covering equivalence, and situates these results within established forcing and definability frameworks, including Mansfield–Solovay-type arguments. The work also outlines substantial open questions about the interplay of continuum size, breadth, and definable coverings in higher dimensions, and discusses potential strengthening via fogs, clouds, and sprays. Overall, it advances a nuanced bridge between descriptive set theory, constructibility, and geometric covering problems with clear directions for future research.

Abstract

Generalizing a result of Törnquist and Weiss, we study the connection between the existence of $\varSigma_2^1$ Sierpiński's coverings of $\mathbb{R}^n$, and a cardinal invariant of the upper semi-lattice of constructibility degrees known as breadth.

The breadth of constructibility degrees and definable Sierpiński's coverings

TL;DR

The paper deepens the link between geometric coverings and set-theoretic structure by showing that the existence of definable Sierpiński coverings is tightly controlled by the breadth of the upper semi-lattice of constructibility degrees. It introduces a real-breadth dichotomy, proves a definable covering equivalence, and situates these results within established forcing and definability frameworks, including Mansfield–Solovay-type arguments. The work also outlines substantial open questions about the interplay of continuum size, breadth, and definable coverings in higher dimensions, and discusses potential strengthening via fogs, clouds, and sprays. Overall, it advances a nuanced bridge between descriptive set theory, constructibility, and geometric covering problems with clear directions for future research.

Abstract

Generalizing a result of Törnquist and Weiss, we study the connection between the existence of Sierpiński's coverings of , and a cardinal invariant of the upper semi-lattice of constructibility degrees known as breadth.
Paper Structure (11 sections, 19 theorems, 37 equations)

This paper contains 11 sections, 19 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sierpiński coverings
  4. Constructibility degrees
  5. Breadth of an upper semi-lattice
  6. The breadth of constructibility degrees
  7. Definable Sierpiński coverings
  8. Open questions
  9. Covering $\mathbb{R} ^2$ with $\varSigma_2^1$ fogs, clouds, and sprays
  10. Large continuum and small breadth of constructibility degrees
  11. Covering $\mathbb{R} ^3$ with constructible continuous functions

Key Result

Lemma 1

Let $n \in \omega$ and suppose that $\mathbb{R}^n = \bigcup_{i < n } A_i$ and Then, the $A_i$s cannot all have the property of Baire.

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Definition 3: Ditor, Ditor:1984aa
  • Theorem 4: Ditor:1984aa
  • Theorem 5: Ditor:1984aa
  • Theorem 6: Wehrung:2010aa
  • Proposition 7
  • Theorem 8: Lévy-Shoenfield Jech:1978aa
  • ...and 22 more