Level of Sets on Spheres

Author: Sonneborn, Lee Myers

Year: 1956

Degree: Dissertation (Ph.D.)

Advisor: Fuller, F. Brock

Committee Member: Unknown, Unknown

Option: Mathematics; Physics

DOI: 10.7907/rjtw-fw15

Abstract

Let f:Sn X I1 → E1 be a continuous, real-valued function on Sn X I1 for > 1. Then for every t Ɛ I1 there is a subset At X t of the n-sphere Sn X t with the following properties:

i) f(At X t) = kt independent of x Ɛ At.

ii) At X t is connected.

iii) (Sn X t) - (At X t) has no component containing more than half the n-dimensional measure of Sn X t.

iv) For any measure-preserving homeomorphism, g, of Sn X t, At X t contains the image of at least one of its points. (e.g. At X t contains a pair of antipodal points of Sn X t)

v) kt varies continuously with t.

Further, if g:T2 E1 is a continuous real-valued function defined on a torus, then there is a connected, non-contractible subset of T2on on which g is constant.

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