Vol. 60, No. 1, 1975

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On tame Cantor sets in spheres having the same projection in each direction

Leslie C. Glaser

Vol. 60 (1975), No. 1, 87–102

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 (n1)-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.

Mathematical Subject Classification
Primary: 57A15
Received: 20 September 1974
Published: 1 September 1975
Leslie C. Glaser