  This note is purely expository. In the course of the Kolmogorov-Arnold
solution of Hilbert's 13th problem on superpositions there appeared the notion
of basic embedding. A subset K of R^2 is basic if for each continuous function
f:K->R there exist continuous functions g,h:R->R such that f(x,y)=g(x)+h(y) for
each point (x,y) in K. We present descriptions of basic subsets of the plane
(with a proof) and description of graphs basically embeddable into the plane
(solutions of Arnold's and Sternfeld's problems). We present some results and
open problems on the smooth version of the property of being basic. This note
is accessible to undergraduates and could be an interesting easy reading for
mature mathematicians. The two sections can be read independently on each
other.
