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Article

Embeddings of finite-dimensional compacta in Euclidean spaces

Topology and its Applications. 2012. No. 159. P. 1670-1677.
Bogatyi S., Valov V., Bogataya S.

If g is a map from a space X into Rm and q is an integer, let B q,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 set of all continuous maps  g : X →Rm such that dim B q,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 dim X ≤ 1.