A simpler proof of Sternfeld’s Theorem

Abstract

In Sternfeld’s work on Kolmogorov’s Superposition Theorem appeared the combinatorial–geometric notion of a basic set and a certain kind of arrays. A subset [Formula: see text] is basic if any continuous function [Formula: see text] could be represented as the sum of compositions of continuous functions [Formula: see text] and projections to the coordinate axes. The definition of a Sternfeld array is presented in this paper. Sternfeld’s Arrays Theorem. If a closed bounded subset[Formula: see text] contains Sternfeld arrays of arbitrary large size then[Formula: see text] is not basic. The paper provides a simpler proof of this theorem.