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Cardinal Characteristics on Bounded Generalised Baire Spaces

Tristan van der Vlugt

TL;DR

The paper extends the framework of cardinal characteristics to bounded generalised Baire spaces with $\kappa$ inaccessible, introducing four families of parametrised invariants on $\prod_{\alpha\in\kappa} b(\alpha)$ and their duals. It develops a robust toolkit of relational systems, Tukey connections, and infimum/supremum operations to compare these invariants, and analyzes triviality vs. nontriviality across parameter choices, highlighting the crucial role of the bound function $h$ and the base $b$. It then establishes a web of provable relations (via Tukey theory) and demonstrates that varying parameters can yield consistently distinct cardinals, using $\,\kappa$-Cohen and $\,\kappa$-Hechler models and Sacks-like forcing to separate invariants on both the $\in^*$ and $\cancel{\ni^\infty}$ sides. The results illuminate how the lack of Lebesgue measure in ${}^{\kappa}\kappa$ interacts with the rich structure of club filters and stationary sets, producing new complexity and open questions for independence and preservation in the generalized setting.

Abstract

We will give an overview of four families of cardinal characteristics defined on subspaces $\prod_{α\inκ}b(α)$ of the generalised Baire space ${}^κκ$, where $κ$ is strongly inaccessible and $b\in{}^κκ$. The considered families are bounded versions of the dominating, eventual difference, localisation and anti-localisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are non-trivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.

Cardinal Characteristics on Bounded Generalised Baire Spaces

TL;DR

The paper extends the framework of cardinal characteristics to bounded generalised Baire spaces with inaccessible, introducing four families of parametrised invariants on and their duals. It develops a robust toolkit of relational systems, Tukey connections, and infimum/supremum operations to compare these invariants, and analyzes triviality vs. nontriviality across parameter choices, highlighting the crucial role of the bound function and the base . It then establishes a web of provable relations (via Tukey theory) and demonstrates that varying parameters can yield consistently distinct cardinals, using -Cohen and -Hechler models and Sacks-like forcing to separate invariants on both the and sides. The results illuminate how the lack of Lebesgue measure in interacts with the rich structure of club filters and stationary sets, producing new complexity and open questions for independence and preservation in the generalized setting.

Abstract

We will give an overview of four families of cardinal characteristics defined on subspaces of the generalised Baire space , where is strongly inaccessible and . The considered families are bounded versions of the dominating, eventual difference, localisation and anti-localisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are non-trivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.
Paper Structure (19 sections, 87 theorems, 18 equations, 4 figures)

This paper contains 19 sections, 87 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation & Definition of Concepts
  3. Bounded Generalised Baire Spaces
  4. Cardinal Characteristics & Relational Systems
  5. Classical and Higher Cichoń Diagram
  6. Trivial Choices of Parameters
  7. Domination & Unboundedness
  8. Localisation & Avoidance
  9. Antilocalisation & Anti-avoidance
  10. Eventual Difference & Cofinal Equality
  11. More Relations Between Cardinals
  12. Relations for Fixed Parameters
  13. Infima and Suprema
  14. Consistency of Strict Inequalities
  15. The $\kappa$-Cohen model
  16. ...and 4 more sections

Key Result

Lemma 2.1

There exists $M\in\mathcal{M}_\kappa^b$ such that the subspace $\prod b\setminus M$ of $\prod b$ is homeomorphic to ${}^{\kappa}\kappa$.

Figures (4)

  • Figure 1: Diagram of the relations between cardinals on bounded spaces
  • Figure 2: The $\kappa$-Cohen model
  • Figure 3: The $\kappa$-Hechler model
  • Figure 4: The dual $\kappa$-Hechler model

Theorems & Definitions (107)

  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 97 more