A description of continua basically embeddable in R2

An embedding K ⊂ Rn is called basic if for every continuous function f:K → R there are continuous functions g1,…,gn:R → R such that for every point (x1,…, xn) ϵ K, f(x1,…,xn) = g1(x1) + … +gn(xn). The problem of describing the compacta basically embeddable in Rn is related to Hilbertʹs 13th problem. The answer for n ≠ 2 was given by Kolmogorov, Arnold, Ostrand and Sternfeld: if K is a compactum of dimension n, then it is basically embeddable in R2n + 1 and (if n ⩾ 2) is not basically embeddable in R2n. The description of pathwise-connected compacta basically embeddable in R2 is given here. Such compacta are dendrites (i.e., acyclic peano continua) containing none of the nine prohibited continua, listed in the paper. The proof is based on Sternfeldʹs reduction of the property of being a basic embedding to a pure geometric condition.