Topology on locally finite metric spaces
Valerio Capraro

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…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research
