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.
Cite
@article{arxiv.1802.08341,
title = {Embeddability on functions: order and chaos},
author = {Raphaël Carroy and Yann Pequignot and Zoltán Vidnyánszky},
journal= {arXiv preprint arXiv:1802.08341},
year = {2024}
}