On projections of a compact set in $\mathbb R^N$

Abstract

We apply ideas of geometric measure theory and Baire category theory to topological problems, namely, to topological embeddings of compact sets into Euclidean spaces. In 1947, Borsuk constructed a Cantor set in $\mathbb R^N$, $N\geqslant 3$, such that its projection onto any $(N-1)$-plane contains an $(N-1)$-dimensional ball. This can be strengthened: a desired Cantor set can be obtained from an arbitrary Cantor set by an arbitrarily small isotopy of the space $\mathbb R^N$. The question arises: how do the dimensions of the projections of a compact set $X\subset \mathbb R^N$ behave under a typical ambient isotopy or under a typical ambient homeomorphism? (Typical in the sense of the Baire category.) We solve this problem. As a consequence, we get new criteria of tameness and wildness of a Cantor set in terms of its projections. Our main result strengthens V{\"a}is\"{a}l\"{a}'s theorem (1979) connecting Hausdorff dimension and Shtan'ko embedding dimension. In its turn, V{\"a}is\"{a}l\"{a}'s theorem extends results of N\"{o}beling (1931) and Szpilrajn (1937) on relationship between Hausdorff dimension and topological dimension.