The maln result of this paper is
that for each n ≧ 2 there exists in En a tame (n − 1)-sphere S containing a twice
tame Cantor subset C such that all projections of the two sets are the same. That is,
if Hα is any (n− 1)-dimensional linear hyperspace in En and πα: En→ Hα denotes
the natural projection of En onto Hα, then for every α we have πα(S) = πα(C). A
number of interesting corollaries follow immediately from this result. One corollary is
that there exists for each n ≧ 2 a countable collection of tame Cantor sets in En such
that each straight line in En intersects a countable number of these Cantor
sets.
