Notre Dame Journal of Formal Logic

Mereology on Topological and Convergence Spaces

Daniel R. Patten

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Abstract

We show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory. We generalize these results to the Cartesian closed category of convergence spaces.

Article information

Source
Notre Dame J. Formal Logic, Volume 54, Number 1 (2013), 21-31.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1355494520

Digital Object Identifier
doi:10.1215/00294527-1731362

Mathematical Reviews number (MathSciNet)
MR3007959

Zentralblatt MATH identifier
1284.03138

Subjects
Primary: 06A06: Partial order, general
Secondary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)

Keywords
mereology topology mereotopology convergence space general extensional mereology set theory

Citation

Patten, Daniel R. Mereology on Topological and Convergence Spaces. Notre Dame J. Formal Logic 54 (2013), no. 1, 21--31. doi:10.1215/00294527-1731362. https://projecteuclid.org/euclid.ndjfl/1355494520


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References

  • [1] Beattie, R., and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Kluwer, Dordrecht, 2002.
  • [2] Binz, E., Continuous Convergence on $C(X)$, vol. 469 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
  • [3] Blair, H. A., D. W. Jakel, R. J. Irwin, and A. Rivera, “Elementary differential calculus on discrete and hybrid structures,” pp. 41–53 in Logical Foundations of Computer Science, vol. 4514 of Lecture Notes in Computer Science, Springer, Berlin, 2007.
  • [4] Bourbaki, N., Elements of Mathematics: General Topology, Part 1, Hermann, Paris, 1966.
  • [5] Casati, R., and A. C. Varzi, Parts and Places: The Structures of Spatial Representation, MIT Press, Cambridge, Mass., 1999.
  • [6] Clarke, B. L., “A calculus of individuals based on ‘connection’,” Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 204–18.
  • [7] Dikranjan, D., and W. Tholen, Categorical Structure of Closure Operators: With Applications to Topology, Algebra and Discrete Mathematics, vol. 346 of Mathematics and its Applications, Kluwer, Dordrecht, 1995.
  • [8] Guarino, N., M. Carrara, and P. Giaretta, “Formalizing ontological commitments,” pp. 560–67 in AAAI ’94: Proceedings of the Twelfth National Conference on Artificial Intelligence, Vol. 1, AAAI Press, Menlo Park, Calif., 1994.
  • [9] Heckmann, R., “A non-topological view of dcpos as convergence spaces,” pp. 159–86 in Topology in Computer Science (Schloß Dagstuhl, Germany, 2000), vol. 305 of Theoretical Computer Science, Elsevier, Amsterdam, 2003.
  • [10] Kelley, J. L., General Topology, D. Van Nostrand, Toronto, 1955.
  • [11] Mac Lane, S., Categories for the Working Mathematician, 2nd edition, vol. 5 of Graduate Texts in Mathematics, Springer, New York, 1998.
  • [12] Mashaal, M., Bourbaki: A Secret Society of Mathematicians, translated from the French original by A. Pierrehumbert, American Mathematical Society, Providence, 2006.
  • [13] Munkres, J. R., Topology: A First Course, Prentice-Hall, Englewood Cliffs, N.J., 1975.
  • [14] Preuß, G., “Semiuniform convergence spaces and filter spaces,” pp. 333–73 in Beyond Topology, vol. 486 of Contemporary Mathematics, American Mathematical Society, Providence, 2009.
  • [15] Randell, D. A., Z. Cui, and A. G. Cohn, “A spatial logic based on regions and connection,” pp. 165–76 in KR 92: Principles of Knowledge Representation and Reasoning (Cambridge, Mass., 1992), Morgan Kaufmann, San Francisco, 1992.