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Finitary substitute for continuous topology

  • Rafael D. Sorkin
Article

Abstract

Finite topological spaces are combinatorial structures that can serve as replacements for, or approximations to, bounded regions within continuous spaces such as manifolds. In this spirit, the present paper studies the approximation of general topological spaces by finite ones, or really by “finitary” ones in case the original space is unbounded. It describes how to associate a finitary spaceF with any locally finite covering of aT1-spaceS; and it shows howF converges toS as the sets of the covering become finer and more numerous. It also explains the equivalent description of finite topological spaces in order-theoretic language, and presents in this connection some examples of posetsF derived from simple spacesS. The finitary spaces considered here should not be confused with the so-called causal sets, but there may be a relation between the two notions in certain situations.

Keywords

Manifold Field Theory Elementary Particle Quantum Field Theory Topological Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Aleksandrov, P. S. (1956).Combinatorial Topology, Greylock, Rochester, New York, Volume I.Google Scholar
  2. Bombelli, L., Lee, J., Meyer, D., and Sorkin, R. D. (1987).Physical Review Letters,59, 521–524, and references therein.Google Scholar
  3. Bourbaki, N. (1966).General Topology, Hermann, Paris, Chapter I, § 1, Problem 2.Google Scholar
  4. Bourbaki, N. (1968).Theory of Sets, Hermann, Paris, Chapter III, § 7.Google Scholar
  5. Duke, D. W., and Owens, I. F., eds. (1985).Advances in Lattice Gauge Theory, World Scientific, Singapore.Google Scholar
  6. Eilenberg, S., and Steenrod, N. (1952).Foundations of Algebraic Topology, Princeton University Press, Princeton, New Jersey.Google Scholar
  7. Finkelstein, D. (1987). Finite physics, inThe Universal Turing Machine—A Half-Century Survey, R. Herken, ed., Kammerer & Unverzagt, Hamburg.Google Scholar
  8. Finkelstein, D. (1988).International Journal of Theoretical Physics,27, 473, and references therein.Google Scholar
  9. Finkelstein, D. (1989a),International journal of Theoretical Physics,28, 441.Google Scholar
  10. Finkelstein, D. (1989b).International Journal of Theoretical Physics,28, 1081.Google Scholar
  11. Finkelstein, D., and Rodriguez, E. (1986).Physica,18D, 197.Google Scholar
  12. Hartle, J. B. and Sorkin, R. D. (1986), unpublished.Google Scholar
  13. Isham, C. J. (1989a). An introduction to general topology and quantum topology, inProceedings of the Advanced Summer Institute on Physics, Geometry and Topology (August 1989).Google Scholar
  14. Isham, C. J. (1989b).Classical and Quantum Gravity,6, 1509.Google Scholar
  15. Kelley, J. L. (1955).General Topology, van Nostrand, Toronto, p. 136.Google Scholar
  16. McCord, M. C. (1966).Duke Mathematics Journal,33, 465.Google Scholar
  17. Sorkin, R. D. (1983). Posets as lattice topologies, inGeneral Relativity and Gravitation, B. Bertotti, F. de Felice, and A. Pascolini, eds., Consiglio Nazionale Delle Ricerche, Rome, Volume I, pp. 635–637.Google Scholar
  18. Sorkin, R. D. (1990). Does a discrete order underly spacetime and its metric?, inProceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics, A. Coley, F. Cooperstock, and B. Tupper, eds., World Scientific, Singapore, and references therein.Google Scholar
  19. Stanley, R. P. (1986).Enumerative Combinatorics, Wadsworth & Brooks/Cole, Monterey, California, Volume I.Google Scholar
  20. Stong, R. E. (1966).Transactions of the American Mathematical Society,123, 325.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Rafael D. Sorkin
    • 1
  1. 1.Institute for Advanced StudyPrinceton

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