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 [9], that for any dense Gδ set G ∈ R 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 later is at all possible, is an open probem.