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The fixed-point property for represented spaces

Mathieu Hoyrup 1
1 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : We investigate which represented spaces enjoy the fixed-point property, which is the property that every continuous multi-valued function has a fixed-point. We study the basic theory of this notion and of its uniform version. We provide a complete characterization of countable-based spaces with the fixed-point property, showing that they are exactly the pointed ω-continuous dcpos. We prove that the spaces whose lattice of open sets enjoys the fixed-point property are exactly the countably-based spaces. While the role played by fixed-point free functions in the diagonal argument is well-known, we show how it can be adapted to fixed-point free multi-valued functions, and apply the technique to identify the base-complexity of the Kleene-Kreisel spaces, which was an open problem.
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https://hal.inria.fr/hal-03117745
Contributor : Mathieu Hoyrup Connect in order to contact the contributor
Submitted on : Thursday, January 28, 2021 - 2:02:54 PM
Last modification on : Saturday, October 16, 2021 - 11:26:10 AM

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  • HAL Id : hal-03117745, version 2

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Mathieu Hoyrup. The fixed-point property for represented spaces. 2021. ⟨hal-03117745v2⟩

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