Topological dimension and sums of connectivity functions

✍ Scribed by Krzysztof Ciesielski; Jerzy Wojciechowski

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 105 KB
• Volume: 112
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00231-x

✦ Synopsis

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 G δ -set G ∈ R 2k+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 latter is at all possible, is an open problem.