Embeddability on functions: order and chaos
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866917806647279616 |
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| author | Carroy, Raphaël Pequignot, Yann Vidnyánszky, Zoltán |
| author_facet | Carroy, Raphaël Pequignot, Yann Vidnyánszky, Zoltán |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_08341 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Embeddability on functions: order and chaos Carroy, Raphaël Pequignot, Yann Vidnyánszky, Zoltán Logic Primary: 03E15, 26A21, 54C05, 54C25, Secondary: 06A07 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. |
| title | Embeddability on functions: order and chaos |
| topic | Logic Primary: 03E15, 26A21, 54C05, 54C25, Secondary: 06A07 |
| url | https://arxiv.org/abs/1802.08341 |