Connectivity functions \(I^ n\to I\) dense in \(I^ n\times I\) (Q910879)

Let X and Y be topological spaces and let \(f: X\to Y\) be a function. Then f is said to be a connectivity function provided that if C is a connected subset of X, then the graph of f restricted to C is a connected subset of \(X\times Y\). Let \(I=[0,1]\) and for a positive integer n let \(\Delta^ n\) denote the unit n-simplex of the Euclidean space \(R^{n+1}\). It is shown that for each \(n\geq 2\) there exist \(2^{{\mathfrak c}}\) connectivity functions \(f: \Delta^ n\to I\) dense in \(\Delta^ n\times I\). As a consequence it is proved that there exists a connectivity function \(f: I^ n\to I^ k\) dense in \(I^ n\times I^ k\) for any \(n\geq 2\) and \(k\geq 2\). The results generalize earlier ones due to \textit{S. K. Hildebrand} and \textit{D. E. Sanderson} [Fundam. Math. 57, 237-245 (1965; Zbl 0149.197)] and to \textit{J. B. Brown} [Colloq. Math. 23, 53-60 (1971; Zbl 0219.54013)].