Topology and Diffeology via Metric-like Functions
Masaki Taho
TL;DR
The paper develops a unified framework for spaces endowed with a family of metric-like functions $d_r$ for $r>0$ to recover topology, uniformity, and a diffeology in a single axiomatic system. By imposing reflexivity, monotonicity in $r$, right-continuity, and a weak triangle inequality, it defines neighborhoods $U_{r,ε}(x)$ and shows these generate a natural topology, potentially non-symmetric, and a corresponding quasi-uniformity. The framework is illustrated on tiling spaces, spaces of submanifolds $Ψ_d(\mathbb{R}^N)$, and graph spaces, showing that the induced tiling topology coincides with the standard one and that the associated diffeologies agree with established constructions (e.g., the MR4882913 diffeology). It also demonstrates how a (non-symmetric) pseudo-metric can arise from $d_r$ and how a diffeology generated by $d_r$ relates to the D-topology, providing a coherent topological–diffeological perspective with potential broad applicability. Altogether, the work offers a practical, axiomatized route to study a broad class of metric-like spaces through both topology and diffeology.
Abstract
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of metric-like functions originally introduced for spaces of submanifolds. We show that the topologies, uniformities, and diffeologies of these spaces can be systematically derived from the proposed axioms. Furthermore, the framework covers examples such as spaces with compact-open topologies, tiling spaces, and spaces of graphs, which have appeared in different contexts. These results support the study of spaces with metric-like structures from both topological and diffeological perspectives.