Limits and sums of extendable connectivity functions (Q1890614)

Let \(R\) be the set of reals and let \(I= [0, 1]\). A function \(G: X\to Y\), between topological spaces \(X\) and \(Y\), is said to be a connectivity function if whenever \(C\) is a connected subset of \(X\), then the graph of the restriction \(G|C: C\to Y\) is a connected subset of \(X\times Y\). A connectivity function \(g: I\to R\) is extendable if there is a connectivity function \(G: I\times I\to R\) for which \(G(x, 0)= g(x)\) when \(0\leq x\leq 1\). Let \(K\) denote a class of functions from \(I\) into \(I\) and let \(g\in K\). A set \(M\subset I\) is said to be \(g\)-negligible with respect to \(K\) if \(f\in K\) whenever \(f: I\to I\) and \(f= g\) on \(I- M\). The main result of the paper reads as follows: Theorem. Suppose \(K\) is the class of extendable connectivity functions and \(g\in K\). Then the following are equivalent: (i) The graph of \(g\) is dense in \(I\times I\); (ii) Every nowhere dense subset of \(I\) is \(g\)-negligible; (iii) There exists a dense \(G_\delta\) subset of \(I\) which is \(g\)-negligible. The author also gives some applications of the above theorem: Each function \(f: I\to I\) is the pointwise limit of a sequence of extendable connectivity functions \(f_n: I\to I\); each function \(f: I\to I\) equals a series \(\sum^\infty_{n= 1} g_n\) of extendable connectivity functions \(g_n: I\to R\).