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A well-quasi-order for continuous functions

Raphaël Carroy, Yann Pequignot

Abstract

We prove that continuous reducibility is a well-quasi-order on the class of continuous functions between separable metrizable spaces with analytic zero-dimensional domain. To achieve this, we define scattered functions, which generalize scattered spaces, and describe exhaustively scattered functions between zero-dimensional separable metrizable spaces up to continuous equivalence.

A well-quasi-order for continuous functions

Abstract

We prove that continuous reducibility is a well-quasi-order on the class of continuous functions between separable metrizable spaces with analytic zero-dimensional domain. To achieve this, we define scattered functions, which generalize scattered spaces, and describe exhaustively scattered functions between zero-dimensional separable metrizable spaces up to continuous equivalence.

Paper Structure

This paper contains 36 sections, 87 theorems, 42 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The context
  3. The strategy
  4. Some known and less known preliminaries
  5. Non-scattered functions
  6. Scattered functions
  7. Disjoint union
  8. Gluing
  9. Gluing as an upper bound
  10. Gluing as a lower bound
  11. Pointed Gluing and the General Structure
  12. Pointed gluing as an upper bound
  13. Maximum functions
  14. Pointed gluing as a lower bound and minimum functions
  15. General Structure
  16. ...and 21 more sections

Key Result

Theorem 1.2

Continuous reducibility is a better-quasi-order on the class of continuous functions $f:X\to Y$ from a zero-dimensional separable metrizable space $X$ to a metrizable space $Y$ such that either $X$ is analytic or $Y$ is countable.

Figures (4)

  • Figure 1: The general structure of continuous reducibility on $\mathsf{Scat}$.
  • Figure 2: The wedge operation ${\bigvee}(f_0,f_1\mid g)$.
  • Figure 3: Proof of the Vertical Theorem.
  • Figure 4: Hasse diagram of our minimal set of generators of $\mathsf{Scat}_{[\lambda,\lambda+1]}$ for $\lambda$ limit or $1$.

Theorems & Definitions (193)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 183 more