Embeddings of finite-dimensional compacta in Euclidean spaces

AbstractIf g is a map from a space X into Rm and q is an integer, let Bq,d,m(g) be the set of all planes Πd⊂Rm such that |g−1(Πd)|⩾q. Let also H(q,d,m,k) denote the maps g:X→Rm such that dimBq,d,m(g)⩽k. We prove that for any n-dimensional metric compactum X each of the sets H(3,1,m,3n+1−m) and H(2,1,m,2n) is dense and Gδ in the function space C(X,Rm) provided m⩾2n+1 (in this case H(3,1,m,3n+1−m) and H(2,1,m,2n) can consist of embeddings). The same is true for the sets H(1,d,m,n+d(m−d))⊂C(X,Rm) if m⩾n+d, and H(4,1,3,0)⊂C(X,R3) if dimX⩽1. This result complements an authorsʼ result from Bogatyi and Valov (2005) [5]. A parametric version of the above theorem, as well as a partial answer of a question from Bogatyi (2008) [4] and Bogatyi and Valov (2005) [5] are also provided