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arXiv · 1802.08341

Embeddability on functions: order and chaos

Raphaël Carroy·Yann Pequignot·Zoltán Vidnyánszky
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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.

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Raphaël Carroy, Yann Pequignot, Zoltán Vidnyánszky. 2018-02-22. Embeddability on functions: order and chaos. https://doi.org/10.1090/tran/7739

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