Sums of Connectivity Functions on Rn


Krzysztof Ciesielski & Jerzy Wojciechowski

Proc. London Math. Soc. 76(2) (1998), 406--426.

A function f from Rn to R is a connectivity function if the graph of its restriction f|C to any connected subset C of Rn is connected. The main goal of the 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 theorem that is of interest on its own. For every dense G-delta subset G of Rn there are homeomorphisms h1,...,hn of Rn such that the sets G, h1(G),...,hn(G) cover Rn.

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Last modified May 3, 1998.