Journal of Symbolic Logic

Dynamic topological logic of metric spaces

David Fernández-Duque

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Abstract

Dynamic Topological Logic (𝒟𝒯ℒ) is a modal framework for reasoning about dynamical systems, that is, pairs 〈X,f〉 where X is a topological space and f: X→ X a continuous function.

In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on ℚ.

We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.

Article information

Source
J. Symbolic Logic, Volume 77, Issue 1 (2012), 308-328.

Dates
First available in Project Euclid: 20 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1327068705

Digital Object Identifier
doi:10.2178/jsl/1327068705

Mathematical Reviews number (MathSciNet)
MR2951643

Zentralblatt MATH identifier
1241.03020

Citation

Fernández-Duque, David. Dynamic topological logic of metric spaces. J. Symbolic Logic 77 (2012), no. 1, 308--328. doi:10.2178/jsl/1327068705. https://projecteuclid.org/euclid.jsl/1327068705


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