All projections of a typical Cantor set are Cantor sets

In 1994, John Cobb asked: given N > m > k > 0, does there exist a Cantor set in RN such that each of its projections into m-planes is exactly k-dimensional? Such sets were described for (N, m, k) = (2, 1, 1) by L. Antoine (1924) and for (N, m, m) by K. Borsuk (1947). Examples were constructed for the cases (3, 2, 1) by J. Cobb (1994), for (N, m, m- 1) and in a different way for (N, N - 1, N - 2) by O. Frolkina (2010, 2019), for (N,N - 1, k) by S. Barov, J.J. Dijkstra and M. van der Meer (2012). We show that such sets are exceptional in the following sense. Let C(R-N) be the set of all Cantor subsets of R-N endowed with the Hausdorff metric. It is known that C(R-N) is a Baire space. We prove that there is a dense G delta subset P subset of C(R-N) such that for each X is an element of P and each non-zero linear subspace L subset of R-N, the orthogonal projection of X into L is a Cantor set. This gives a partial answer to another question of J. Cobb stated in the same paper (1994). (C) 2020 Elsevier B.V. All rights reserved.