The fixed-point property for represented spaces

Mathieu Hoyrup (1)

1 MOCQUA - Designing the Future of Computational Models

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.

Domaines: Logique en informatique [cs.LO]; Topologie générale [math.GN]

Mathieu Hoyrup. The fixed-point property for represented spaces. 2021. ⟨hal-03117745v1⟩