Topology on locally finite metric spaces

TL;DR

This paper develops a new topological framework for locally finite metric spaces, introducing NPP-based notions and invariants, including a fundamental group, with applications in group amenability, graph theory, and metric inequalities.

Contribution

It introduces NPP-function, NPP-homotopy, and NPP-isomorphism concepts, along with invariants like the fundamental group, extending classical topology to locally finite metric spaces.

Findings

Defined NPP-homotopy and NPP-isomorphism.

Constructed invariants including the fundamental group.

Extended metric inequalities and topological theorems to locally finite spaces.

Abstract

The necessity of a theory of General Topology and, most of all, of Algebraic Topology on locally finite metric spaces comes from many areas of research in both Applied and Pure Mathematics: Molecular Biology, Mathematical Chemistry, Computer Science, Topological Graph Theory and Metric Geometry. In this paper we propose the basic notions of such a theory and some applications: we replace the classical notions of continuous function, homeomorphism and homotopic equivalence with the notions of NPP-function, NPP-local-isomorphism and NPP-homotopy (NPP stands for Nearest Point Preserving); we also introduce the notion of NPP-isomorphism. We construct three invariants under NPP-isomorphisms and, in particular, we define the fundamental group of a locally finite metric space. As first applications, we propose the following: motivated by the longstanding question whether there is a purely metric condition which extends the notion of amenability of a group to any metric space, we propose the property SN (Small Neighborhood); motivated by some applicative problems in Computer Science, we prove the analog of the Jordan curve theorem in $\mathbb Z^2$; motivated by a question asked during a lecture at Lausanne, we extend to any locally finite metric space a recent inequality of P.N.Jolissaint and Valette regarding the $\ell_p$-distortion.