### Abstract

We study the quasi-order of topological embeddability on definable functions between Polish 0-dimensional spaces. We consider 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 wellquasi- order. We also investigate the existence of maximal elements with respect to embeddability in a given Baire class. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.

Original language | English |
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Pages (from-to) | 6711-6738 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 9 |

DOIs | |

Publication status | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*371*(9), 6711-6738. https://doi.org/10.1090/tran/7739