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Metrizability and Coconnectedness

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Abstract

Solving the problem stated in Sichler and Trnková, Topol. Its Appl., 142: 159–179, 2004, we construct metrics μ, ν on a set P such that the spaces X=(P,μ) and Y=(P,ν) have the same monoid of all continuous selfmaps, the space Y is coconnected (in the sense that every continuous map Y×Y→Y depends on at most one coordinate) while X is not. Also, properties of the forgetful functors Metr → Unif → Top are investigated for the “simultaneous variant” of the above problem.

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References

  1. Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley, New York (1990) (see online version http://katmat.math.uni-bremen.de/acc)

  2. Cook, H.: Continua which admit only the identity mapping onto non-degenerate sub-continua. Fund. Math. 60, 241–249 (1967)

    MATH  MathSciNet  Google Scholar 

  3. Engelking, R.: General Topology, Państwowe Wydawnictvo Naukowe, Warszawa 1977, 2nd edn. (1985)

  4. Herrlich, H.: On the concept of reflections in general topology. In: Proceeding Symp. on Extension Theory of Topological Structures, Berlin (1967)

  5. Herrlich, H.: Topologische Reflexionen und Coreflexionen. In: Lect. Notes in Math., vol. 78. Springer, Berlin Heidelberg New York (1968)

    Google Scholar 

  6. Kuratowski, C.: Topologie I, II, Monografie Matematyczne, Warsaw (1950)

  7. Lawvere, F.W.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. 50, 869–872 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lawvere, F.W.: Some algebraic problems in context of functorial semantics of algebraic theories. In: Lect. Notes in Math., vol. 61, pp. 41–46. Springer, Berlin Heidelberg New York (1968)

    Google Scholar 

  9. McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. 1. Brooks/Cole, Monterey, CA (1978)

    Google Scholar 

  10. Pultr, A., Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North Holland, Amsterdam (1980)

    MATH  Google Scholar 

  11. Sichler, J., Trnková, V.: Maps between a space and its square. Topology Appl. 142, 159–179 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Szendrei, Á.: Clones in Universal Algebra. Press de l’Université de Montréal (1986)

  13. Taylor, W.: The Clone of a Topological Space, Research and Exposition Math., vol. 13, Helderman-Verlag, Berlin (1986)

    Google Scholar 

  14. Taylor, W.: Abstract clone theory. In: Rosenberg, I.G., Sabidussi, G. (eds.) Algebras and Order, Proceedings of the NATO Advanced Study Institute, Montréal 1991, pp. 507–530. Kluwer, Boston, MA (1993)

    Google Scholar 

  15. Trnková, V.: Semirigid spaces. Trans. Amer. Math. Soc. 343, 305–329 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Trnková, V.: Co-connected spaces. Serdica Math. J. 24, 25–36 (1998)

    MATH  MathSciNet  Google Scholar 

  17. Trnková, V.: Clones, coclones and coconnected spaces. Acta Math. Univ. Comenian. (N.S.) 69, 241–259 (2000)

    MATH  MathSciNet  Google Scholar 

  18. Trnková, V.: Counting cocomponents of a topological space. Appl. Categ. Structures 12, 379–396 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute of Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Věra Trnková

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  1. Věra Trnková
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Correspondence to Věra Trnková.

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Supported by the Grant Agency of Czech Republic under grant 201/06/0664 and by the project of Ministry of Education of Czech Republic MSM 0021620839.

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Trnková, V. Metrizability and Coconnectedness. Appl Categor Struct 15, 621–631 (2007). https://doi.org/10.1007/s10485-006-9036-5

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  • DOI: https://doi.org/10.1007/s10485-006-9036-5

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