Intuitionistic overlap structures

Francesco Ciraulo



We study some connections between two kinds of \emph{overlap} relations: that of point-free geometries in the sense of Grzegorczyk, Whitehead and Clarke, and that recently introduced by Sambin within his constructive approach to topology. The main thesis of this paper is that the overlap relation in the latter sense is a necessary tool for a constructive and intuitionistic development of point-free geometry.


overlap algebras; connection structures; mereological fields; constructive reasoning

Full Text:



Biacino, L., and G. Gerla, “Connection structures”, Notre Dame J. Formal Logic, 32 (1991): 242–247.

Biacino, L., and G. Gerla, “Connection structures: Grzegorczyk’s and Whitehead’s definiitons of point”, Notre Dame J. Formal Logic, 37 (1996): 431–439.

Ciraulo, F., “Regular opens in formal topology and a representation theorem for overlap algebras”, Ann. Pure Appl. Logic, 164 (2013): 421–436.

Ciraulo, F., M.E. Maietti and P. Toto, “Constructive version of Boolean algebra”, Logic Journal of The IJPL, 21 (2013): 44–62.

Ciraulo, F., and G. Sambin, “The overlap algebra of regular opens”, J. Pure Appl. Algebra, 214 (2010): 1988–1995.

Gerla, G., “Pointless geometries”, in Handbook of Incidence Geometry, North-Holland, Amsterdam, 1995, pp. 1015–1031.

Grzegorczyk, A., “Axiomatizability of geometry without points”, Synthese, 12 (1960): 228–235.

Johnstone, P.T., Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1986.

Joyal, A., and M. Tierney, An extension of the Galois Theory of Grothendieck, Memoirs of the Amer. Math. Soc. 309, 1984.

Mac Lane, S., and I. Moerdijk, Sheaves in Geometry and Logic. A First Introduction to Topos Theory, Springer-Verlag, New York, 1994.

Sambin, G., “Some points in formal topology”, Theoretical Computer Science, 305 (2003): 347–408.

Sambin, G., The Basic Picture and Positive Topology. New structures for Constructive Mathematics, Oxford University Press, Oxford (to appear).

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism