A notion of continuity in discrete spaces and applications
This work addresses foundational issues in discrete mathematics and metric geometry, offering theoretical tools for analyzing continuity and distortion in discrete settings, which is incremental but extends prior results to more general contexts.
The paper tackles the problem of defining continuity in discrete spaces by proposing a notion of continuous paths for locally finite metric spaces, leading to results such as an analogue of the Jordan curve theorem in Z^2 and an extension of an inequality for ℓ^p-distortion to a broad class of spaces, including all finite metric spaces.
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 Z^2 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 \ell^p-distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.