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


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 everyxF. 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.


Continuous Function Real Number Topological Space Arbitrary Function Finite Subset 
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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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