Embeddability on functions: order and chaos

TL;DR

This paper investigates the complexity and structure of embeddability among definable functions between Polish zero-dimensional spaces, revealing a dichotomy in complexity and the uniqueness of maximal elements in the class of continuous functions.

Contribution

It establishes a dichotomy in the descriptive complexity of the embeddability quasi-order and characterizes the existence of maximal elements across Baire classes.

Findings

Embeddability quasi-order is either analytic complete or a well-quasi-order.

Only continuous functions have a maximal element in their Baire class.

No other Baire class admits a maximal element, unlike continuous functions which have a maximum.

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.