Hostname: page-component-6766d58669-vgfm9 Total loading time: 0 Render date: 2026-05-21T13:33:23.042Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatiality for formal topologies

Published online by Cambridge University Press:  01 February 2007

NICOLA GAMBINO  and
PETER SCHUSTER
NICOLA GAMBINO
Affiliation:
Laboratoire de Combinatoire et Informatique Mathématique, Université du Québec à Montréal, Case Postale 8888, Succ. Centre-Ville, Montréal (Québec) H3C 3P8, Canada Email: gambino@math.uqam.ca
PETER SCHUSTER
Affiliation:
Mathematisches Institut, Universität München, Theresienstrasse 39, 80333 Müunchen, Germany Email: Peter.Schuster@mathematik.uni-munchen.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 1 , February 2007 , pp. 65 - 80
Copyright
Copyright © Cambridge University Press 2007

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.)

Article purchase

Temporarily unavailable

References

Aczel, P. (1986) The type theoretic interpretation of Constructive Set Theory: inductive definitions. In: Marcus, R. B., Dorn, G. and Weingartner, P. (eds.) Logic, Methodology and Philosophy of Science VII, North-Holland 17–49.Google Scholar
Aczel, P. (2006) Aspects of general topology in constructive set theory. Ann. Pure Appl. Logic 137 (1–3)3–29.CrossRefGoogle Scholar
Aczel, P. and Rathjen, M. (2001) Notes on constructive set theory. Technical Report 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences.Google Scholar
Banaschewski, B. (1983) The power of the ultrafilter theorem. J. London Math. Soc. 27 193–202.CrossRefGoogle Scholar
Battillotti, G. and Sambin, G. (2006) Pretopologies and a uniform presentation of sup-lattices, quantales and frames. Ann. Pure Appl. Logic 137 30–61.Google Scholar
Coquand, T., Sambin, G., Smith, J. and Valentini, S. (2003) Inductively generated formal topologies. Ann. Pure Appl. Logic 124 (1-3)71–106.CrossRefGoogle Scholar
Curi, G. (2003) Geometry of observations, Ph.D. thesis, Università di Siena.Google Scholar
Curi, G. (2006) On the collection of points of a formal space. Ann. Pure Appl. Logic 137 (1–3)126–146.Google Scholar
Dummett, M. (2000) Elements of Intuitionism, Oxford University Press, second edition.Google Scholar
Fourman, M. and Grayson, R. (1982) Formal spaces. In: Troelstra, A. S. and van Dalen, D. (eds.) The L.E.J. Brouwer Centenary Symposium, North-Holland 107–122.Google Scholar
Fourman, M. P. and Hyland, J. M. E. (1979) Sheaf models for analysis. In: Fourman, M. P., Mulvey, C. J. and Scott, D. S. (eds.) Applications of Sheaves. Springer-Verlag Lecture Notes in Mathematics 753280–301.CrossRefGoogle Scholar
Fourman, M. P. and Scott, D. S. (1979) Sheaves and logic. In: Fourman, M. P., Mulvey, C. J. and Scott, D. S. (eds.) Applications of Sheaves. Springer-Verlag Lecture Notes in Mathematics 753302–401.CrossRefGoogle Scholar
Fox, C. (2005) Point-set and point-free topology in constructive set theory, Ph.D. thesis, Department of Mathematics, University of Manchester.Google Scholar
Gambino, N. (2002) Sheaf interpretations for generalised predicative intuitionistic systems, Ph.D. thesis, Department of Computer Science, University of Manchester.Google Scholar
Gambino, N. (2006) Heyting-valued interpretations for constructive set theory. Ann. Pure Appl. Logic 137 (1–3)164–188.CrossRefGoogle Scholar
Gambino, N. and Aczel, P. (2006) The generalised type-theoretic interpretation of Constructive Set Theory. J. Symb. Logic 71 (1)63–103.CrossRefGoogle Scholar
Hochster, M. (1969) Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 43–60.CrossRefGoogle Scholar
Johnstone, P. T. (1977) Rings, fields, and spectra. J. Algebra 49 283–260.Google Scholar
Johnstone, P. T. (1982) Stone Spaces, Cambridge University Press.Google Scholar
Johnstone, P. T. (1983) The point of pointless topology. Bull. Amer. Math. Soc. 8 (8)41–53.CrossRefGoogle Scholar
Johnstone, P. T. (1991) The art of pointless thinking: a student's guide to the category of locales. In: Herrlich, H. and Porst, H.-E. (eds.) Category Theory at Work, Heldermann Verlag 85–107.Google Scholar
Joyal, A. and Tierney, M. (1984) An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. 390.Google Scholar
Mac Lane, S. and Moerdijk, I. (1994) Sheaves in Geometry and Logic, Springer-Verlag.CrossRefGoogle Scholar
Maietti, M. E. and Sambin, G. (2005) Towards a minimalistic foundation for constructive mathematics. In: Crosilla, L. and Schuster, P. (eds.) From Sets and Types to Topology and Analysis, Oxford University Press 91–114.CrossRefGoogle Scholar
Maietti, M. E. and Valentini, S. (2004) A structural investigation on formal topology: coreflection of formal covers and exponentiability. J. Symb. Logic 69 (4)967–1005.Google Scholar
Martin-Löf, P. (1984) Intuitionistic Type Theory – Notes by G. Sambin of a course given in Padua in 1980, Bibliopolis.Google Scholar
Moerdijk, I. (1984) Heine-Borel does not imply the fan theorem. J. Symb. Logic 49 (2)514–519.Google Scholar
Myhill, J. (1975) Constructive Set Theory. J. Symb. Logic 40 (3)347–382.CrossRefGoogle Scholar
Negri, S. (2002) Continuous domains as formal spaces. Mathematical Structures in Computer Science 12 19–52.CrossRefGoogle Scholar
Nordström, B., Petersson, K. and Smith, J. M. (2000) Martin-Löf Type Theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science 5, Oxford University Press.Google Scholar
Palmgren, E. (2005a) From intuitionistic to point-free topology. U.U.D.M. Report 2005:47, Department of Mathematics, University of Uppsala.Google Scholar
Palmgren, E. (2005b) Quotient spaces and coequalisers in formal topology. J. UCS 11 (12)1996–2007.Google Scholar
Rav, Y. (1977) Variants of Rado's selection lemma and their applications. Math. Nachr. 79 145–165.CrossRefGoogle Scholar
Sambin, G. (1987) Intuitionistic formal spaces – a first communication. In: Skordev, D. (ed.) Mathematical Logic and its applications, Plenum 187–204.Google Scholar
Sambin, G. (2003) Some points in formal topology. Theoretical Computer Science 305 347–408.Google Scholar
Sambin, G. and Gebellato, S. (1999) A preview of the basic picture: a new perspective on formal topology. In: Altenkirch, T., Naraschewski, W. and Reus, B. (eds.) Types for proofs and programs (Irsee 1997). Springer-Verlag Lecture Notes in Computer Science 1657194–207.CrossRefGoogle Scholar
Schuster, P. (2006) Formal Zariski topology: positivity and points. Ann. Pure Appl. Logic 137 (1–3)317–359.CrossRefGoogle Scholar
Sigstam, I. (1995) Formal spaces and their effective presentations. Arch. Math. Logic 34 (4)211–246.CrossRefGoogle Scholar
Simmons, H. (1978) A framework for topology. In: MacIntyre, A., Pacholski, L. and Paris, J. (eds.) Logic Colloquium '77, North-Holland 239–251.CrossRefGoogle Scholar
Simmons, H. (2004) The coverage technique for enriched posets. Available from the author's web page.Google Scholar
Vickers, S. (1989) Topology via Logic, Cambridge University Press.Google Scholar