Journal of Soviet Mathematics

, Volume 50, Issue 2, pp 1497–1512 | Cite as

On pointwise approximation of arbitrary functions by countable families of continuous functions

  • A. B. Arkhangel'skii
  • D. B. Shakhmatov
Article
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Abstract

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F⊂X and every real number ε>0 one can choosen∈ω such that ∥f(x)−fn(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A⊂X, there exists a continuous mapfA:X→Rω such that A=f A −1 (A). Splitting spaces will be studied systematically.

Keywords

Continuous Function Real Number Topological Space Arbitrary Function Finite Subset 
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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Literature cited

  1. 1.
    A. V. Arkhangel'skii and D. B. Shakhmatov, “Splitting spaces and questions of the approximation of functions,” Fifth Tiraspol Symposium on General Topology and Its Applications [in Russian], Shtiintsa, Kishinev (1985), pp. 10–11.Google Scholar
  2. 2.
    A. V. Arkhangel'skii and D. B. Shakhmatov, “Splitting spaces,” Scientific-Research Seminar on General Topology (Spring Semester Sesions, Academic Year 1984/85) [in Russian], Vestnik Moskov. Univ., Ser. Mat. Mekh., No. 5, 90 (1985).Google Scholar
  3. 3.
    A. V. Arkhangel'skii and V. I. Ponomarev, Fundamentals of General Topology in Problems and Exercises [in Russian], Nauka, Moscow (1974).Google Scholar
  4. 4.
    R. Engelking, General Topology, PWN, Warsaw (1977).Google Scholar
  5. 5.
    J. L. Kelley, General Topology [Russian translation], Nauka, Moscow (1981).Google Scholar
  6. 6.
    A. V. Arkhangel'skii, “Structure and classification of topological spaces and cardinal invariants,” Uspekhi Mat. Nauk,33, No. 6, 29–84 (1978).Google Scholar
  7. 7.
    A. V. Arkhangel'skii, “On a class of spaces containing all metric and all locally bicompact spaces,” Dokl. Akad. Nauk SSSR,151, No. 4, 751–754 (1963).Google Scholar
  8. 8.
    A. V. Arkhangel'skii, “On a class of spaces containing all metric and all locally bicompact spaces,” Mat. Sbornik,67, No. 1, 55–88 (1965).Google Scholar
  9. 9.
    A. Okuyama, “On metrizability of M-spaces,” Proc. Japan. Acad.,40, 176–179 (1964).Google Scholar
  10. 10.
    C. J. E. Borges, “Stratifiable spaces,” Pacif. J. Math.,17, 1–16 (1966).Google Scholar
  11. 11.
    M. M. Choban, “Some metrization theorems for pinnate spaces,” Dokl. Akad. Nauk SSSR,173, No. 6, 1270–1272 (1967).Google Scholar
  12. 12.
    M. M. Choban and N. K. Dodon, Theory ofP-Scattered Spaces [in Russian], Shtiintsa, Kishinev (1979).Google Scholar
  13. 13.
    K. Nagami, “Σ-Spaces,” Fund. Math.,65, 160–192 (1969).Google Scholar
  14. 14.
    N. Noble, “The density character of function spaces,” Proc. Amer. Math. Soc.,42, No. 1, 228–233 (1974).Google Scholar
  15. 15.
    A. V. Arkhangel'skii, “The general concept of splitting of topological spaces over a class of spaces,” Fifth Tiraspol Symposium on General Topology and Its Applications [in Russian], Shtiintsa, Kishinev (1985), pp. 8–10.Google Scholar
  16. 16.
    A. V. Arkhangel'skii, “Some new directions in the theory of continuous maps,” in: Continuous Functions on Topological Spaces [in Russian], Riga (1986), pp. 5–35.Google Scholar
  17. 17.
    V. V. Tkacuk, “Approximation of Rx with countable subsets of Cp(X) and calibers of the space Cp(X),” Comment. Math. Univ. Carolinae,27, No. 2, 267–276 (1986).Google Scholar
  18. 18.
    H. J. Kowalsky, “Einbettung metrischer Räume,” Arch. der Math.,8, 336–339 (1957).Google Scholar
  19. 19.
    I. Juhâsz, “On extremal values of mappings, I,” Annales Univ. Sci. Budapest Sectia Math.,6, 39–42 (1963).Google Scholar
  20. 20.
    N. V. Velichko, “A note on pinnate spaces,” Czechoslovak Math. J.,25, No. 1, 8–19 (1975).Google Scholar
  21. 21.
    A. V. Arkhangel'skii, “Function spaces in pointwise convergence topology and compact spaces,” Uspekhi Mat. Nauk,39, No. 5, 11–50 (1984).Google Scholar
  22. 22.
    L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, N.J. (1960).Google Scholar
  23. 23.
    H. J. K. Junnila, “Stratifiable preimages of topological spaces,” Colloq. Math. Soc. Janos Bolyai,23, Topology, Budapest (1978), pp. 689–703.Google Scholar
  24. 24.
    A. V. Arkhangel'skii, “Classes of topological groups,” Uspekhi Mat. Nauk,36, No. 3, 127–146 (1981).Google Scholar
  25. 25.
    A. V. Arkhangel'skii, “Factorization theorems and function spaces: stability and monolithicity,” Dokl. Akad. Nauk SSSR,265, No. 5, 1039–1043 (1982).Google Scholar
  26. 26.
    V. V. Uspenskii, “On the spectrum of frequencies of function spaces,” Vestnik Mosk. Univ., Ser. Mat. Mekh., No. 1, 35–37 (1982).Google Scholar
  27. 27.
    M. E. Gewand, “The Lindelöf degrees of scattered spaces and their products,” J. Austral. Math. Soc., Ser. A,37, No. 1, 98–105 (1984).Google Scholar
  28. 28.
    E. G. Pytkeev, “On hereditarily pinnate spaces,” Mat. Zametki,28, No. 4, 603–618 (1980).Google Scholar
  29. 29.
    Z. Balogh, “On the metrizability of Fpp-spaces and its relationship to the normal Moore space conjecture,” Fund. Math.,113, No. 1, 45–58 (1981).Google Scholar
  30. 30.
    B. A. Pasynkov, “On open countably-multiply maps,” in: Seminar on General Topology [in Russian], Izd. Moskov. Univ., Moscow (1981), pp. 85–114.Google Scholar
  31. 31.
    E. Michael, “Bi-quotient maps and Cartesian products of quotient maps,” Ann. Inst. Fourier,18, No. 2, 287–302 (1969).Google Scholar
  32. 32.
    E. Michael, “A quintuple quotient quest,” General Topology Appl.,2, 91–138 (1972).Google Scholar
  33. 33.
    A. V. Arkhangel'skii, “On invariants of the type of character and weight,” Trudy Moskov. Mat. Obshch.,38, 3–27 (1979).Google Scholar
  34. 34.
    A. V. Arkhangel'skii, “Spectrum of frequencies of topological spaces and the product operation,” Trudy Moskov. Mat. Obshch.,40, 171–206 (1979).Google Scholar
  35. 35.
    Sh. Kakutani, “Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Tokyo,12, 82–84 (1936).Google Scholar
  36. 36.
    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. I, Springer, Berlin (1963).Google Scholar
  37. 37.
    G. Birkhoff, “A note on topological groups,” Compositio Math.,3, 427–430 (1936).Google Scholar
  38. 38.
    D. B. Shakhmatov, “No upper bound for cardinalities of Tychonoff c.c.c. spaces when a Gδ-diagonal exists (An answer to J. Ginsburg and R. G. Woods' question),” Comment. Math. Univ. Carolinae,25, No. 4, 731–746 (1984).Google Scholar
  39. 39.
    B. A. Pasynkov, “Almost metrizable topological groups,” Dokl. Akad. Nauk SSSR, 161, No. 281–284 (1965).Google Scholar
  40. 40.
    A. A. Markov, “On free topological groups,” Dokl. Akad. Nauk SSSR,31, No. 4, 299–301 (1941).Google Scholar
  41. 41.
    A. A. Markov, “On free topological groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,9, 3–64 (1945).Google Scholar
  42. 42.
    A. V. Arkhangel'skii, “On relations between invariants of topological groups and their subspaces,” Uspekhi Mat. Nauk,35, No. 3, 3–22 (1980).Google Scholar
  43. 43.
    W. Comfort and D. Grant, “Cardinal invariants, pseudocompactness and minimality: some recent advances in the topological theory of topological groups,” Topol. Proc.,6, 227–265 (1981).Google Scholar
  44. 44.
    D. B. Shakhmatov, “On pseudocompact spaces with pointwise-countable base,” Dokl. Akad. Nauk SSSR,279, No. 4, 825–829 (1984).Google Scholar
  45. 45.
    F. Siwiec, “On defining a space by a weak base,” Pacif. J. Math.,52, 233–243 (1974).Google Scholar
  46. 46.
    V. V. Uspenskii, “A large Fσ discrete Fréchet space having the Suslin property,” Comment. Math. Univ. Carolinae,25, No. 2, 257–260 (1984).Google Scholar
  47. 47.
    A. Okuyama, “A survey of the theory of σ-spaces,” General Topol. Appl.,1, No. 1, 57–63 (1971).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • A. B. Arkhangel'skii
  • D. B. Shakhmatov

There are no affiliations available

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