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Cardinal characteristics associated with small subsets of reals

Miguel A. Cardona, Adam Marton, Jaroslav Supina

Abstract

Inspired by Bartoszyński's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszyński's and Shelah's representation of $F_σ$ measure zero sets. We show they are $σ$-ideals characterizing all small sets and $F_σ$ measure zero sets. We also study the cardinal characteristics associated with the introduced ideals. We use them to describe the invariants of measure, discuss their connection to Cichoń's diagram, and present related consistency results.

Cardinal characteristics associated with small subsets of reals

Abstract

Inspired by Bartoszyński's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszyński's and Shelah's representation of measure zero sets. We show they are -ideals characterizing all small sets and measure zero sets. We also study the cardinal characteristics associated with the introduced ideals. We use them to describe the invariants of measure, discuss their connection to Cichoń's diagram, and present related consistency results.
Paper Structure (7 sections, 40 theorems, 84 equations, 2 figures)

This paper contains 7 sections, 40 theorems, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Layering of small sets
  3. Monotonicity
  4. Additivity and cofinality of the new ideals
  5. Covering and uniformity of the new ideals
  6. Consistency results
  7. Open Questions

Key Result

Theorem 1

$\nu_i=\kappa_i$ for $i<2$, where

Figures (2)

  • Figure 1: Cichoń's diagram including the cardinal characteristics associated with our ideals, and $\hbox{\rm add}(\mathcal{E}) = \hbox{\rm add}(\mathcal{M})$ and $\hbox{\rm cof}(\mathcal{E}) = \hbox{\rm cof}(\mathcal{M})$ due to Bartoszyński and Shelah BS1992. The relations hold for contributive pairs.
  • Figure 2: Inclusions.

Theorems & Definitions (92)

  • Definition 1.1: bartosmallBART2018
  • Theorem 1: \ref{['thm:alocsmall']}
  • Theorem 2: \ref{['cov_non']}, \ref{['Thm:cofadd']}
  • Theorem 1.3: BS1992
  • Theorem 3: \ref{['a11']}
  • Theorem 4: \ref{['NAbound']}
  • Theorem 5: \ref{['SEcomparison']}
  • Theorem 6
  • Definition 2.1
  • Lemma 2.2
  • ...and 82 more