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{ä}isälä's theorem (1979) connecting Hausdorff dimension and Shtan'ko embedding dimension. In its turn, V{ä}isälä's theorem extends results of Nöbeling (1931) and Szpilrajn (1937) on relationship between Hausdorff dimension and topological dimension.
From: Olga Frolkina [view email]
[v1]
Sat, 30 May 2026 09:29:34 UTC (17 KB)
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