Special embeddings of finite-dimensional compacta in Euclidean spaces

摘要

If g is a map from a space X into R~m and z (∈) g(X), let P2.1,m(g.z) be the set of all lines ∏~1(∪)R~m containing z such that |g~(-1)(∏~1)|≥2. We prove that for any n-dimensional metric compactum X the functions g : X R~m, where m ≥2n + 1. with dim P2.1.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.