Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-11T10:24:04.114Z Has data issue: false hasContentIssue false

Tychonoff's theorem in the framework of formal topologies

Published online by Cambridge University Press:  12 March 2014

Sara Negri
Affiliation:
Dipartimento di Matematica Pura Ed Applicata, Università di Padova, Via G. Belzoni N.7, I–35131 Padova, Italy E-mail: negri@math.unipd.it
Silvio Valentini
Affiliation:
Dipartimento di Matematica Pura Ed Applicata, Università di Padova, Via G. Belzoni N.7, I–35131 Padova, Italy E-mail: silvio@math.unipd.it

Extract

In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]).

The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.

Type
Research Article
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]Aczel, P., An introduction to inductive definitions, Handbook of mathematical logic (Barwise, J., editor), North-Holland, 1977, pp. 739782.CrossRefGoogle Scholar
[2]Backhouse, R., Chisholm, P., Malcom, G., and Saaman, E., Do-it-yourself type theory, Formal Aspects of Computing, vol. 1 (1989), pp. 1984.CrossRefGoogle Scholar
[3]Battilotti, G. and Sambin, G., Pretopologies and a uniform presentation of sup-lattices, quantales and frames, manuscript, 1993.Google Scholar
[4]Cederquist, J., A machine assisted formalization of pointfree topology in type theory, Licentiate thesis, Chalmers University of Technology and University of Göteborg, Sweden, 1994.Google Scholar
[5]Cederquist, J., Coquand, T., and Negri, S., Helly-Hahn-Banach informal topology, Twenty-five years of constructive type theory (Smith, J. and Sambin, G., editors), Oxford University Press, in press.Google Scholar
[6]Cederquist, J. and Negri, S., A constructive proof of the Heine-Borel covering theorem for formal reals, Types for proofs and programs (Berardi, S. and Coppa, M., editors), Lecture Notes in Computer Science, vol. 1158, Springer-Verlag, 1996, pp. 6275.CrossRefGoogle Scholar
[7]Coquand, T., An intuitionistic proof of Tychonoff's theorem, this Journal, vol. 57 (1992), no. 1, pp. 2832.Google Scholar
[8]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, Amsterdam, 1982, pp. 107122.Google Scholar
[9]Johnstone, P. T., Tychonoff's theorem without the axiom of choice, Fundamenta Mathematicae, vol. 113 (1981), pp. 2135.CrossRefGoogle Scholar
[10]Johnstone, P. T., Stone spaces, Cambridge University Press, 1982.Google Scholar
[11]Johnstone, P. T., The point of pointless topology, Bulletin of the American Mathematical Society, vol. 8 (1983), pp. 4143.CrossRefGoogle Scholar
[12]Johnstone, P. T. and Vickers, S., Preframe presentations present, Category theory (Proceedings, Como 1990) (Pedicchio, Carboni, and Rosolini, , editors), Lecture Notes in Mathematics, vol. 1488, 1991, pp. 193212.Google Scholar
[13]Kelley, J. L., The Tychonoff product theorem implies the axiom of choice, Fundamenta Mathematicae, vol. 37 (1950), pp. 7576.CrossRefGoogle Scholar
[14]Kriz, I., A constructive proof of Tychonoffs theorem for locales, Commentationes Mathematicae Universitatis Carolinae, vol. 26 (1985), no. 3, pp. 619630.Google Scholar
[15]Magnusson, L., The implementation of ALF—a proof editor based on Martin-Löf's monomorphic type theory with explicit substitution, Ph.D. thesis, Chalmers University of Technology and University of Göteborg, 1995.Google Scholar
[16]Martin-Löf, P., Intuitionistic type theory, notes by Sambin, Giovanni of a series of lectures given in Padua, June 1980, Bibliopolis, Naples, 1984.Google Scholar
[17]Moerdijk, I., Continuous fibrations and inverse limits of toposes, Compositio Mathematica, vol. 58 (1986), pp. 4572.Google Scholar
[18]Negri, S., Dalla topologia formate all'analisi, Ph.D. thesis, University of Padua, Department of Mathematics, 1996.Google Scholar
[19]Nordström, B., Petersson, K., and Smith, J., Programming in Martin-Löf's type theory, Oxford University Press, 1990.Google Scholar
[20]Sambin, G., Intuitionistic formal spaces—a first communication, Mathematical logic and its applications (Skordev, D., editor), Plenum, 1987, pp. 187204.CrossRefGoogle Scholar
[21]Sambin, G., Intuitionistic formal spaces and their neighborhood, Logic Colloquium 88 (Ferro, , Bonotto, , Valentini, , and Zanardo, , editors), North-Holland, 1989, pp. 261285.Google Scholar
[22]Sambin, G. and Valentini, S., Building up a tool-box for Martin-Löf intuitionistic type theory, Twenty-five years of constructive type theory (Smith, J. and Sambin, G., editors), Oxford University Press, in press.CrossRefGoogle Scholar
[23]Sambin, G., Valentini, S., and Virgili, P., Constructive domain theory as a branch of intuitionistic pointfree topology, Theoretical Computer Science, vol. 159 (1996), pp. 319341.CrossRefGoogle Scholar
[24]Valentini, S., Decidability in intuitionistic theory of types is functionally decidable, Mathematical Logic Quarterly, vol. 42 (1996), pp. 300304.CrossRefGoogle Scholar
[25]Venetus, Paulus (Sambin, Alias G., Valentini, S.), Propositum cameriniense sive etiam itinera certaminis inter rationem insiemes aedificandi analisym situs intuitionisticamque et mathematicae artium reliquas res, Atti degli incontri di logica matematica (Gerla, , Toffalori, , and Tulipani, , editors), vol. 8, 1993, pp. 115143.Google Scholar
[26]Vermeulen, J. J. C., Constructive techniques in functional analysis, Ph.D. thesis, University of Sussex, 1986.Google Scholar
[27]Vermeulen, J. J. C., Proper maps of locales, Journal of Pure and Applied Algebra, vol. 92 (1994), pp. 79107.CrossRefGoogle Scholar
[28]Vickers, S., Topology via logic, Cambridge Tracts in Theoretical Computer Science, vol. 5, Cambridge University Press, 1989.Google Scholar