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Minimal invariant spaces in formal topology

Published online by Cambridge University Press:  12 March 2014

Thierry Coquand
Thierry Coquand*
Affiliation:
Department of Computer Science, Chalmers University, S 41296, Göteborg, Sweden, E-mail: coquand@cs.chalmers.se
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Extract

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).

In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 689 - 698
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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