标题： 基于类似度量函数的拓扑与微分几何 显示英文标题

标题： Topology and Diffeology via Metric-like Functions

摘要： 本文研究了一类配备满足特定公理的类似度量函数的空间。这些函数为定义拓扑、一致性和微分结构提供了一个统一框架。该框架基于最初为子流形空间引入的一族类似度量函数。我们证明了这些空间的拓扑、一致性和微分结构可以系统地从所提出的公理推导出来。此外，该框架涵盖了紧开拓扑空间、平铺空间和图空间等例子，这些例子出现在不同的背景下。这些结果支持从拓扑和微分视角对具有类似度量结构的空间进行研究。 显示英文摘要

摘要： 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.