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Topological dimension and sums of connectivity functions☆☆ This work was partially supported by NSF Cooperative Research Grant INT-9600548 with its Polish part being financed by Polish Academy of Science PAN.

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
112
Issue
2
Identifiers
DOI: 10.1016/s0166-8641(99)00231-x
Keywords
  • Inductive Dimension
  • Connectivity Functions
  • Darboux Functions
Disciplines
  • Mathematics

Abstract

Abstract 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 indX or indX−1 . Whether the latter is at all possible, is an open problem.

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