A description of continua basically embeddable in [formula omitted

Abstract

An embedding K ⊂ R n is called basic if for every continuous function f:K → R there are continuous functions g 1,…,g n: R → R such that for every point ( x 1,…, x n ) ϵ K, f( x 1,…, x n ) = g 1( x 1) + … + g n ( x n ). The problem of describing the compacta basically embeddable in R n is related to Hilbert's 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 R 2n + 1 and (if n ⩾ 2) is not basically embeddable in R 2n . The description of pathwise-connected compacta basically embeddable in R 2 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 Sternfeld's reduction of the property of being a basic embedding to a pure geometric condition.