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Spatiality for formal topologies

  • NICOLA GAMBINO (a1) and PETER SCHUSTER (a2)

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.

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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 1749.
Aczel, P. (2006) Aspects of general topology in constructive set theory. Ann. Pure Appl. Logic 137 (1–3)329.
Aczel, P. and Rathjen, M. (2001) Notes on constructive set theory. Technical Report 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences.
Banaschewski, B. (1983) The power of the ultrafilter theorem. J. London Math. Soc. 27 193202.
Battillotti, G. and Sambin, G. (2006) Pretopologies and a uniform presentation of sup-lattices, quantales and frames. Ann. Pure Appl. Logic 137 3061.
Coquand, T., Sambin, G., Smith, J. and Valentini, S. (2003) Inductively generated formal topologies. Ann. Pure Appl. Logic 124 (1-3)71106.
Curi, G. (2003) Geometry of observations, Ph.D. thesis, Università di Siena.
Curi, G. (2006) On the collection of points of a formal space. Ann. Pure Appl. Logic 137 (1–3)126146.
Dummett, M. (2000) Elements of Intuitionism, Oxford University Press, second edition.
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 107122.
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 753280301.
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 753302401.
Fox, C. (2005) Point-set and point-free topology in constructive set theory, Ph.D. thesis, Department of Mathematics, University of Manchester.
Gambino, N. (2002) Sheaf interpretations for generalised predicative intuitionistic systems, Ph.D. thesis, Department of Computer Science, University of Manchester.
Gambino, N. (2006) Heyting-valued interpretations for constructive set theory. Ann. Pure Appl. Logic 137 (1–3)164188.
Gambino, N. and Aczel, P. (2006) The generalised type-theoretic interpretation of Constructive Set Theory. J. Symb. Logic 71 (1)63103.
Hochster, M. (1969) Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 4360.
Johnstone, P. T. (1977) Rings, fields, and spectra. J. Algebra 49 283–260.
Johnstone, P. T. (1982) Stone Spaces, Cambridge University Press.
Johnstone, P. T. (1983) The point of pointless topology. Bull. Amer. Math. Soc. 8 (8)4153.
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 85107.
Joyal, A. and Tierney, M. (1984) An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. 390.
Mac Lane, S. and Moerdijk, I. (1994) Sheaves in Geometry and Logic, Springer-Verlag.
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 91114.
Maietti, M. E. and Valentini, S. (2004) A structural investigation on formal topology: coreflection of formal covers and exponentiability. J. Symb. Logic 69 (4)9671005.
Martin-Löf, P. (1984) Intuitionistic Type Theory – Notes by G. Sambin of a course given in Padua in 1980, Bibliopolis.
Moerdijk, I. (1984) Heine-Borel does not imply the fan theorem. J. Symb. Logic 49 (2)514519.
Myhill, J. (1975) Constructive Set Theory. J. Symb. Logic 40 (3)347382.
Negri, S. (2002) Continuous domains as formal spaces. Mathematical Structures in Computer Science 12 1952.
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.
Palmgren, E. (2005a) From intuitionistic to point-free topology. U.U.D.M. Report 2005:47, Department of Mathematics, University of Uppsala.
Palmgren, E. (2005b) Quotient spaces and coequalisers in formal topology. J. UCS 11 (12)19962007.
Rav, Y. (1977) Variants of Rado's selection lemma and their applications. Math. Nachr. 79 145165.
Sambin, G. (1987) Intuitionistic formal spaces – a first communication. In: Skordev, D. (ed.) Mathematical Logic and its applications, Plenum 187204.
Sambin, G. (2003) Some points in formal topology. Theoretical Computer Science 305 347408.
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 1657194207.
Schuster, P. (2006) Formal Zariski topology: positivity and points. Ann. Pure Appl. Logic 137 (1–3)317359.
Sigstam, I. (1995) Formal spaces and their effective presentations. Arch. Math. Logic 34 (4)211246.
Simmons, H. (1978) A framework for topology. In: MacIntyre, A., Pacholski, L. and Paris, J. (eds.) Logic Colloquium '77, North-Holland 239251.
Simmons, H. (2004) The coverage technique for enriched posets. Available from the author's web page.
Vickers, S. (1989) Topology via Logic, Cambridge University Press.

Spatiality for formal topologies

  • NICOLA GAMBINO (a1) and PETER SCHUSTER (a2)

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