All projections of a typical Cantor set are Cantor sets

Most Cantor sets in $\mathbb R^N$ are in general position with respect to all projections

“…As a consequence, we get the following result. Its first part was proved in [8] by a straightforward geometric reasoning. Corollary 3.5.…”

“…In the same paper, Cobb posed another question [4, p. 128]: "Cantor sets that raise dimension under all projections and those in general position with respect to all projections are both dense in the Cantor sets in R mwhich (if either) is more common, in the sense of category or dimension or anything?" In [8], I answered a weaker version of this question showing that all projections of a typical Cantor set in Euclidean space are Cantor sets; thus the examples listed above are "a rarity". In this article, we fully answer the category part of this question: a typical Cantor set in R N is in general position with respect to all projections.…”

Most Cantor sets in $\mathbb R^N$ are in general position with respect to all projections

“…As a consequence, we get the following result. Its first part was proved in [8] by a straightforward geometric reasoning. Corollary 3.4.…”

“…In 1906, L. Zoretti recalls as a known fact ("ce resultat bien connu etant acquis...") that the graph of any such surjection is a Cantor set in plane whose projection to y-axis coincides with I. (See [7] or [8] for historic details. )…”