Special embeddings of finite-dimensional compacta in Euclidean spaces

If g is a map from a space X into Rm and z∉g(X), let P2,1,m(g,z) be the set of all lines Π1⊂Rm containing z such that |g−1(Π1)|⩾2. We prove that for any n-dimensional metric compactum X the functions g:X→Rm, where m⩾2n+1, with for all z∉g(X) form a dense Gδ-subset of the function space C(X,Rm). A parametric version of the above theorem is also provided.