Special embeddings of finite-dimensional compacta in Euclidean spaces

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 ⩾ 2 n + 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.