A notion of continuity in discrete spaces and applications

Valerio Capraro

Abstract

We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z2 of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the ℓp-distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.

Keywords

A-homotopy theory; ℓp-distortion; Digital Jordan curve theorem

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References

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1. Discrete homology theory for metric spaces
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Bulletin of the London Mathematical Society  vol: 46  issue: 5  first page: 889  year: 2014  
doi: 10.1112/blms/bdu043



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