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Infinite trees

Alexandre Goy

TL;DR

The paper develops a descriptive-set-theoretic framework for infinite trees by formalizing languages, prefixes, and word operations, and then building a rigorous topological treatment of trees via Cantor-Bendixson derivatives. It establishes a deep correspondence between closed subsets of $\Sigma^\omega$ and pruned trees, providing a constructive way to pass between branch sets and tree representations using the prefix operator and off-words. The Cantor-Bendixson derivative and its transfinite iterations yield the Cantor-Bendixson rank for both spaces and trees, with results clarifying when trees are thin and how rank behaves under branch decompositions. The work highlights how the topology of infinite branches interacts with tree operations, yielding insights into the structure and cardinality of branches across finite and infinite alphabets, and providing canonical pruning that aligns with topological closedness.

Abstract

A few notes about infinite trees in a descriptive set-theoretic setting.

Infinite trees

TL;DR

The paper develops a descriptive-set-theoretic framework for infinite trees by formalizing languages, prefixes, and word operations, and then building a rigorous topological treatment of trees via Cantor-Bendixson derivatives. It establishes a deep correspondence between closed subsets of and pruned trees, providing a constructive way to pass between branch sets and tree representations using the prefix operator and off-words. The Cantor-Bendixson derivative and its transfinite iterations yield the Cantor-Bendixson rank for both spaces and trees, with results clarifying when trees are thin and how rank behaves under branch decompositions. The work highlights how the topology of infinite branches interacts with tree operations, yielding insights into the structure and cardinality of branches across finite and infinite alphabets, and providing canonical pruning that aligns with topological closedness.

Abstract

A few notes about infinite trees in a descriptive set-theoretic setting.
Paper Structure (19 sections, 29 theorems, 39 equations)

This paper contains 19 sections, 29 theorems, 39 equations.

Table of Contents

  1. Basics on Languages
  2. Concatenation.
  3. Restriction and Co-Restriction.
  4. Prefixing.
  5. Basics on Trees
  6. Concatenation and Infinite Branches.
  7. Graphical Representations.
  8. Construction and Destruction of Trees
  9. Around the Root
  10. Along a Branch
  11. The Cantor-Bendixson Derivative
  12. Derivating Topological Spaces
  13. Derivating Trees
  14. Topologies on Trees
  15. Steady Convergence.
  16. ...and 4 more sections

Key Result

Proposition 1

For all $u, v \in \Sigma^\infty$ and $L \in P(\Sigma^\infty)$

Theorems & Definitions (69)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 3: The binary alphabet
  • Definition 4: Tree
  • Proposition 5
  • proof
  • Definition 6
  • Proposition 7
  • ...and 59 more