Topology and Diffeology via Metric-like Functions

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.