Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-11T09:09:48.834Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Boileau, A. and Joyal, A., La logique des topos, this Journal, vol. 46, pp. 6–16.Google Scholar
[2]Coquand, Th., A formal space of ultrafilter, 1994, manuscript.Google Scholar
[3]Coquand, Th., A constructive topological proof of van der Waerden's theorem, Journal of Pure and Applied Algebra, vol. 105 (1995), pp. 251–259.CrossRefGoogle Scholar
[4]de Bruijn, N. and van der Meiden, W., Notes on Gelfand's theory, Indigationes, vol. 31 (1968), pp. 467–474.Google Scholar
[5]Fourman, M. P. and Grayson, R. J., Formal spaces, The L. E. J. Brouwer centenary symposium (Troelstra, A. S. and van Dalen, D., editors), North-Holland, 1982, pp. 107–122.Google Scholar
[6]Furstenberg, H. and Weiss, B., Topological dynamics and combinatorial number theory, Journal D'Analyse Mathématique, vol. 34 (1978), pp. 61–85.CrossRefGoogle Scholar
[7]Girard, J. Y., Proof-theory and logical complexity, Bibliopolis, 1987.Google Scholar
[8]Graham, R., Rothschild, B., and Spencer, J., Ramsey theory, John Wiley & Sons, New York, 1980.Google Scholar
[9]Hilbert, D., Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88 (1923), pp. 151–165.CrossRefGoogle Scholar
[10]Johnstone, P., Stone spaces, Cambridge Studies in Advanced Mathematics, 1981.Google Scholar
[11]Lorenzen, P. and Myhill, J., Constructive definition of certain sets of numbers, this Journal, vol. 24 (1959), pp. 37–49.Google Scholar
[12]Martin-Löf, P., Notes on constructive mathematics, Almqvist and Wiskell, Stockholm, 1970.Google Scholar
[13]Moerdijk, I. and Wraith, G., Connected locally connected toposes are path-connected, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 849–859.CrossRefGoogle Scholar
[14]Mulvey, Ch. and Pelletier, J. W., A globalization of the Hahn-Banach theorem, Advances in Mathematics, vol. 89 (1991), pp. 1–59.CrossRefGoogle Scholar
[15]Sambin, G., Pretopologies and completeness proofs, this Journal, vol. 60 (1995), pp. 863–878.Google Scholar
[16]van der Meiden, W., Point-free carrier space topology for commutative Banach algebra, Ph.D. thesis, Eindhoven, 1967.Google Scholar