  This note is purely expository. We show how in the course of the
Kolmogorov-Arnold solution of Hilbert's 13-th problem on superpositions there
appeared the notion of a basic embedding. A subset K of R^2 is {\it 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 and 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.
