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Sums of Connectivity Functions on Rn
by
Jurek Wojciechowski
University of West Virginia University
A function f: Rn --> R is a connectivity function if the graph of its restriction f|C to any connected C subset Rn is connected in Rn ×Rn. The main goal of this paper is to prove that every function f: Rn --> R is a sum of n+1 connectivity functions. We will also show that if n > 1, then every function g: Rn --> R which is a sum of n connectivity functions is continuous on some perfect set which implies that the number n+1 in our theorem is the best possible.
To prove the above results, we establish and then apply the following theorems that are of interest on their own.
For every dense G\delta subset G of [0, 1]n there are homeomorphisms h1, ... hn of [0, 1]n such that [0, 1]n = G \cup h1(G) \cup ... hn(G).
For every n > 1 and any connectivity function f: Rn --> R, if x in Rn and \epsilon > 0 then there exists an open set U subset Rn such that x in U subset Bn(x, \epsilon), f|bd(U) is continuous and |f(x) - f(y)| < \epsilon for every y in bd(U).
Date received: March 21, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caac-18.