All projections of a typical Cantor set are Cantor sets

Abstract

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(RN) be the set of all Cantor subsets of RN endowed with the Hausdorff metric. It is known that C(RN) is a Baire space. We prove that there is a dense Gδ subset P⊂C(RN) such that for each X∈P and each non-zero linear subspace L⊂RN, 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).