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A constructive Galois connection between closure and interior

Published online by Cambridge University Press:  12 March 2014

Francesco Ciraulo  and
Giovanni Sambin
Francesco Ciraulo
Affiliation:
Dipartimento di Matematica, Università di Padova, Via Trieste, 63 1-35121 Padova, Italy, E-mail: ciraulo@math.unipd.it
Giovanni Sambin
Affiliation:
Dipartimento di Matematica, Università di Padova, Via Trieste, 63 1-35121 Padova, Italy, E-mail: sambin@math.unipd.it
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Abstract

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 4 , December 2012 , pp. 1308 - 1324
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1]Aczel, P. and Fox, C., Separation properties in constructive topology, From sets and types to topology and analysis: towards practicable foundations for constructive mathematics (Crosilla, L. and Schuster, P., editors), Oxford Logic Guides, vol. 48, Oxford U. P., 2005, pp. 176–192.CrossRefGoogle Scholar
[2]Birkhoff, G., Lattice theory, Colloquium Publications, vol. 25, American Mathematical Society, 1940.Google Scholar
[3]Ciraulo, F., Regular opens in formal topology and a representation theorem for overlap algebras, Annals of Pure and Applied Logic, to appear.Google Scholar
[4]Ciraulo, F. and Sambin, G., The overlap algebra of regular opens, Journal of Pure and Applied Algebra, vol. 214 (2010), pp. 1988–1995.CrossRefGoogle Scholar
[5]Coquand, T., Sadocco, S., Sambin, G., and Smith, J., Formal topologies on the set of first-order formulae, this Journal, vol. 65 (2000), pp. 1183–1192.Google Scholar
[6]Coquand, T., Sambin, G., Smith, J., and Valentini, S., Inductively generated formal topologies, Annals of Pure and Applied Logic, vol. 124 (2003), pp. 71–106.CrossRefGoogle Scholar
[7]Fourman, M. and Scott, D., Sheaves and logic, Applications ofsheaves. Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 21, 1977 (Fourman, M., Mulvey, C., and Scott, D., editors), Lecture Notes in Mathematics, vol. 753, Springer, 1979.Google Scholar
[8]Grayson, R.J., On closed subsets of the intuitionistic reals, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 29 (1983), pp. 7–9.CrossRefGoogle Scholar
[9]Johnstone, P.T., Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, 1983.Google Scholar
[10]Jónsson, B. and Tarski, A., Boolean algebras with operators. I, American Journal of Mathematics, vol. 73 (1951), pp. 891–939.CrossRefGoogle Scholar
[11]Joyal, A. and Tierney, M., An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematical Society, vol. 51 (1984).CrossRefGoogle Scholar
[12]Lane, S. Mac, Categories for the working mathematician, second ed., Springer, 1998.Google Scholar
[13]Martin-Löf, P. and Sambin, G., Generatingpositivity by coinduction, in [15].Google Scholar
[14]Sambin, G., Some points in formal topology, Theoretical Computer Science, vol. 305 (2003), pp. 347–408.CrossRefGoogle Scholar
[15]Sambin, G., The basic picture and positive topology. New structures for constructive mathematics, Oxford University Press, to appear.Google Scholar