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The Spherical Two-Piece Property and Tight Surfaces in Spheres

by Thomas F. Banchoff

A set, A, in E^3, is said to have the *spherical two-piece
property* (STPP) if no plane or sphere in E^3 separtes A into more
than two pieces. Spheres, planes, circles, and tori all have the
STPP. The first part of this paper gives a characterization
of all closed sets in the plane that have the STPP.

A surface, M^2, in E^n is said to be *tight* if no hyperplane
separtes M^2 into more than two pieces. This paper identifies
all tight smooth surfaces that are subsets of spheres in E^n. The
main result in this paper in that the only tight surfaces in
S^3 besides S^2 are the images under inverse stereographic
projection of cyclides of Dupin, a class of alegraic tori
in E^3.