Special embeddings of finite-dimensional compacta in Euclidean spaces

Abstract

If g is a map from a space X into R m and z ∈ g(X), let P 2,1,m (g, z) be the set of all lines Π 1 ⊂ R m containing z such that |g −1 (Π 1 )| ≥ 2. We prove that for any n-dimensional metric compactum X the functions g : X → R m , where m ≥ 2n + 1, with dim P 2,1,m (g, z) ≤ 0 for all z ∈ g(X) form a dense G δ -subset of the function space C(X, R m ). A parametric version of the above theorem is also provided.