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This note is purely expository. We show how in the course of theKolmogorov-Arnold solution of Hilbert's 13-th problem on superpositions thereappeared the notion of a basic embedding. A subset K of R^2 is {\it basic} iffor each continuous function f:K->R there exist continuous functions g,h:R->Rsuch that f(x,y) = g(x) + h(y) for each point (x,y) in K. We presentdescriptions of basic subsets of the plane and graphs basically embeddable intothe plane (solutions of Arnold's and Sternfeld's problems). We present someresults and open problems on the smooth version of the property of being basic.This note is accessible to undergraduates and could be an interesting easyreading for mature mathematicians.

Basic embeddings and Hilbert's 13th problem on superpositions (in Russian)