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 aT 1-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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aleksandrov, P. S. (1956).Combinatorial Topology, Greylock, Rochester, New York, Volume I.
Bombelli, L., Lee, J., Meyer, D., and Sorkin, R. D. (1987).Physical Review Letters,59, 521–524, and references therein.
Bourbaki, N. (1966).General Topology, Hermann, Paris, Chapter I, § 1, Problem 2.
Bourbaki, N. (1968).Theory of Sets, Hermann, Paris, Chapter III, § 7.
Duke, D. W., and Owens, I. F., eds. (1985).Advances in Lattice Gauge Theory, World Scientific, Singapore.
Eilenberg, S., and Steenrod, N. (1952).Foundations of Algebraic Topology, Princeton University Press, Princeton, New Jersey.
Finkelstein, D. (1987). Finite physics, inThe Universal Turing Machine—A Half-Century Survey, R. Herken, ed., Kammerer & Unverzagt, Hamburg.
Finkelstein, D. (1988).International Journal of Theoretical Physics,27, 473, and references therein.
Finkelstein, D. (1989a),International journal of Theoretical Physics,28, 441.
Finkelstein, D. (1989b).International Journal of Theoretical Physics,28, 1081.
Finkelstein, D., and Rodriguez, E. (1986).Physica,18D, 197.
Hartle, J. B. and Sorkin, R. D. (1986), unpublished.
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).
Isham, C. J. (1989b).Classical and Quantum Gravity,6, 1509.
Kelley, J. L. (1955).General Topology, van Nostrand, Toronto, p. 136.
McCord, M. C. (1966).Duke Mathematics Journal,33, 465.
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.
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.
Stanley, R. P. (1986).Enumerative Combinatorics, Wadsworth & Brooks/Cole, Monterey, California, Volume I.
Stong, R. E. (1966).Transactions of the American Mathematical Society,123, 325.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sorkin, R.D. Finitary substitute for continuous topology. Int J Theor Phys 30, 923–947 (1991). https://doi.org/10.1007/BF00673986
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00673986