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

Article

## Abstract

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable family*f*_{ 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 choose*n*∈ω such that ∥f(x)−f_{n}(x)∥<ε for every*x*∈*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 mapf_{A}:*X→R*^{ω} such that A=f _{A} ^{−1} (A). Splitting spaces will be studied systematically.

## Keywords

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

- 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.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.A. V. Arkhangel'skii and V. I. Ponomarev, Fundamentals of General Topology in Problems and Exercises [in Russian], Nauka, Moscow (1974).Google Scholar
- 4.R. Engelking, General Topology, PWN, Warsaw (1977).Google Scholar
- 5.J. L. Kelley, General Topology [Russian translation], Nauka, Moscow (1981).Google Scholar
- 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.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.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.A. Okuyama, “On metrizability of M-spaces,” Proc. Japan. Acad.,40, 176–179 (1964).Google Scholar
- 10.C. J. E. Borges, “Stratifiable spaces,” Pacif. J. Math.,17, 1–16 (1966).Google Scholar
- 11.M. M. Choban, “Some metrization theorems for pinnate spaces,” Dokl. Akad. Nauk SSSR,173, No. 6, 1270–1272 (1967).Google Scholar
- 12.M. M. Choban and N. K. Dodon, Theory ofP-Scattered Spaces [in Russian], Shtiintsa, Kishinev (1979).Google Scholar
- 13.K. Nagami, “Σ-Spaces,” Fund. Math.,65, 160–192 (1969).Google Scholar
- 14.N. Noble, “The density character of function spaces,” Proc. Amer. Math. Soc.,42, No. 1, 228–233 (1974).Google Scholar
- 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.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.V. V. Tkacuk, “Approximation of R
^{x}with countable subsets of C_{p}(X) and calibers of the space C_{p}(X),” Comment. Math. Univ. Carolinae,27, No. 2, 267–276 (1986).Google Scholar - 18.H. J. Kowalsky, “Einbettung metrischer Räume,” Arch. der Math.,8, 336–339 (1957).Google Scholar
- 19.I. Juhâsz, “On extremal values of mappings, I,” Annales Univ. Sci. Budapest Sectia Math.,6, 39–42 (1963).Google Scholar
- 20.N. V. Velichko, “A note on pinnate spaces,” Czechoslovak Math. J.,25, No. 1, 8–19 (1975).Google Scholar
- 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.L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, N.J. (1960).Google Scholar
- 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.A. V. Arkhangel'skii, “Classes of topological groups,” Uspekhi Mat. Nauk,36, No. 3, 127–146 (1981).Google Scholar
- 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.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.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.E. G. Pytkeev, “On hereditarily pinnate spaces,” Mat. Zametki,28, No. 4, 603–618 (1980).Google Scholar
- 29.Z. Balogh, “On the metrizability of F
_{pp}-spaces and its relationship to the normal Moore space conjecture,” Fund. Math.,113, No. 1, 45–58 (1981).Google Scholar - 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.E. Michael, “Bi-quotient maps and Cartesian products of quotient maps,” Ann. Inst. Fourier,18, No. 2, 287–302 (1969).Google Scholar
- 32.E. Michael, “A quintuple quotient quest,” General Topology Appl.,2, 91–138 (1972).Google Scholar
- 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.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.Sh. Kakutani, “Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Tokyo,12, 82–84 (1936).Google Scholar
- 36.E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. I, Springer, Berlin (1963).Google Scholar
- 37.G. Birkhoff, “A note on topological groups,” Compositio Math.,3, 427–430 (1936).Google Scholar
- 38.D. B. Shakhmatov, “No upper bound for cardinalities of Tychonoff c.c.c. spaces when a G
_{δ}-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.B. A. Pasynkov, “Almost metrizable topological groups,” Dokl. Akad. Nauk SSSR, 161, No. 281–284 (1965).Google Scholar
- 40.A. A. Markov, “On free topological groups,” Dokl. Akad. Nauk SSSR,31, No. 4, 299–301 (1941).Google Scholar
- 41.A. A. Markov, “On free topological groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,9, 3–64 (1945).Google Scholar
- 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.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.D. B. Shakhmatov, “On pseudocompact spaces with pointwise-countable base,” Dokl. Akad. Nauk SSSR,279, No. 4, 825–829 (1984).Google Scholar
- 45.F. Siwiec, “On defining a space by a weak base,” Pacif. J. Math.,52, 233–243 (1974).Google Scholar
- 46.V. V. Uspenskii, “A large F
_{σ}discrete Fréchet space having the Suslin property,” Comment. Math. Univ. Carolinae,25, No. 2, 257–260 (1984).Google Scholar - 47.A. Okuyama, “A survey of the theory of σ-spaces,” General Topol. Appl.,1, No. 1, 57–63 (1971).Google Scholar

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