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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

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 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.

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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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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