Topological dimension and sums of connectivity functions

The main goal of this paper is to show that the inductive dimension of a σ-compact metric space X can be characterized in terms of algebraical sums of connectivity (or Darboux) functions X→R. As an intermediate step we show, using a result of Hayashi [Topology Appl. 37 (1990) 83], that for any dense Gδ-set G∈R2k+1 the union of G and some k homeomorphic images of G is universal for k-dimensional separable metric spaces. We will also discuss how our definition works with respect to other classes of Darboux-like functions. In particular, we show that for the class of peripherally continuous functions on an arbitrary separable metric space X our parameter is equal to either indX or indX−1. Whether the latter is at all possible, is an open problem.