Topological Dimension and Sums of Connectivity Functions


Krzysztof Ciesielski & Jerzy Wojciechowski

Topology Appl. 112(2) (2001), 181-191.

The main goal of this paper is to show that the inductive dimension of a sigma-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, that for any dense G\delta set G in 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 ind(X) or ind(X-1). Whether the later is at all possible, is an open probem.

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Last modified March 16, 2001.