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Rough Sets pp 213-250 | Cite as

Topological Structures

  • Lech Polkowski
Part of the Advances in Soft Computing book series (AINSC, volume 15)

Abstract

Topology is a theory of certain set structures which have been motivated by attempts to generalize geometric reasoning and replace it by more flexible schemes. In many schemes of reasoning one resorts to the idea of a neighbor with the assumption that reasonably selected neighbors of a given object preserve its properties in satisfactory degree (cf. methods based on the notion of the nearest neighbor.) The notion of a neighbor as well as a more general notion of a neighborhood are studied by topology.

Keywords

Topological Space Boolean Algebra Topological Structure Compact Space Open Ball 
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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Works quoted

  1. [Alexandrov25]
    P. S. Alexandrov, Zur Begründung der n-dimensionalen mengentheoretischen Topologie, Math. Ann., 94 (1925), pp. 296–308.MathSciNetCrossRefGoogle Scholar
  2. [Alexandrov—Urysohn29]
    P. S. Alexandrov and P. S. Urysohn, Mémoire sur les espaces topologiques compacts, Verh. Konink. Akad. Amsterdam, 14(1929).Google Scholar
  3. [Baire899]
    R. Baire, Ann. di Math., 3(1899).Google Scholar
  4. [Banach22]
    S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), pp. 133–181.MATHGoogle Scholar
  5. [Cantor883]
    G. Cantor, Math. Ann., 21(1883). [Cantor880] G. Cantor, Math. Ann, 17(1880).Google Scholar
  6. [Čech66]
    E. Čech, Topologicke’ prostory, in: E. Cech, Topological Spaces, Academia, Praha, 1966.Google Scholar
  7. [Fréchet28]
    M. Fréchet, Les espaces abstraits, Paris, 1928.Google Scholar
  8. [Fréchet06]
    M. Fréchet, Sur quelques points du Calcul fonctionnel, Rend. Circ. Matem. di Palermo, 22(1906).Google Scholar
  9. [Hahn32]
    H. Hahn, Reelle Funktionen I, Leipzig, 1932.Google Scholar
  10. [Hausdorffl4]
    F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914.Google Scholar
  11. [Kuratowski22]
    C. Kuratowski, Sur l’operation A de l’Analysis Situs, Fund. Math., 3 (1922), pp. 182–199.MATHGoogle Scholar
  12. [Lebesgue05]
    H. Lebesgue, J. de Math., 6(1905).Google Scholar
  13. [Marcus94]
    S. Marcus, Tolerance rough sets, Cech topologies, learning processes, Bull. Polish Acad. Sci. Tech., 42 (1994), pp. 471–487.MATHGoogle Scholar
  14. [Polkowski—Skowron—Zytkow94]
    L. Polkowski, A. Skowron, and J. Zytkow, Tolerance based rough sets, in: T. Y. Lin and M. Wildberger (eds.), Soft Computing: Rough Sets, Fuzzy Logic, Neural Networks, Uncertainty Management, Knowledge Discovery, Simulation Councils, Inc., San Diego, 1995, pp. 55–58.Google Scholar
  15. [Pompéju05]
    D. Pompéju, Ann. de Toulouse, 7(1905).Google Scholar
  16. [Rasiowa—Sikorski50]
    H. Rasiowa and R. Sikorski, A proof of the completeness theorem of Gödel, Fund. Math., 37 (1950), pp. 193–200.MathSciNetMATHGoogle Scholar
  17. [Riesz09]
    F. Riesz, Stetigskeitbegriff und abstrakte Mengenlehre, Atti IV Congr. Int. Mat., Rome, 1909.Google Scholar
  18. [Stone36]
    M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc., 40 (1936), pp. 37–111.Google Scholar
  19. [Tikhonov35]
    A. N. Tikhonov, Über einen Funktionenraum, Math. Ann., 111(1935).Google Scholar
  20. [Vietoris2l]
    L. Vietoris, Monat. Math. Ph., 31 (1921), pp. 173–204.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lech Polkowski
    • 1
    • 2
  1. 1.Polish-Japanese Institute of Information TechnologyWarsawPoland
  2. 2.Department of Mathematics and Computer ScienceUniversity of Wormia and MazuryOlsztynPoland

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