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Embeddability on functions: order and chaos

Raphaël Carroy, Yann Pequignot, Zoltán Vidnyánszky

Abstract

We study the quasi-order of topological embeddability on definable functions between Polish zero-dimensional spaces. We first study the descriptive complexity of this quasi-order restricted to the space of continuous functions. Our main result is the following dichotomy: the embeddability quasi-order restricted to continuous functions from a given compact space to another is either an analytic complete quasi-order or a well-quasi-order. We then turn to the existence of maximal elements with respect to embeddability in a given Baire class. It is proved that the class of continuous functions is the only Baire class to admit a maximal element. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.

Embeddability on functions: order and chaos

Abstract

We study the quasi-order of topological embeddability on definable functions between Polish zero-dimensional spaces. We first study the descriptive complexity of this quasi-order restricted to the space of continuous functions. Our main result is the following dichotomy: the embeddability quasi-order restricted to continuous functions from a given compact space to another is either an analytic complete quasi-order or a well-quasi-order. We then turn to the existence of maximal elements with respect to embeddability in a given Baire class. It is proved that the class of continuous functions is the only Baire class to admit a maximal element. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.

Paper Structure

This paper contains 19 sections, 36 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces
  4. Quasi-orders
  5. Some Cantor--Bendixson analysis
  6. Embedding functions
  7. Let's embed them!
  8. A limitation result on possible topologies for continuous functions
  9. Chaos
  10. One special case, or maybe two
  11. The general case
  12. Order
  13. For a fistful of better-quasi-orders
  14. For a few well-quasi-orders more
  15. Maximal elements in Baire classes
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ and $Y$ be Polish $0\text{-}$dimen-sional spaces. If $X$ has infinitely many non-isolated points and $Y$ is not discrete, then there exists a continuous map $\mathbb{G} \to C(X,Y)$ which reduces $\preccurlyeq$ to $\sqsubseteq$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2: Louveau--Rosendal
  • Proposition 2.4
  • proof
  • Definition 3.1
  • ...and 68 more