AI 中文总结
研究配备权重函数的拓扑空间仅近似满足某性质时能否恢复原始结构，通过给开集赋权研究单调性等近似版本，得出赫尔斯 - 乌拉姆型稳定性结果，涵盖特定拓扑情形并给出相关结果。
AI 中文摘要
本文核心问题是：配备权重函数的拓扑空间仅近似满足某性质时，能否在不大幅改变权重的情况下恢复原始结构？给拓扑空间的每个开集赋予非负权重，研究单调性、次可加性和凸性的近似版本，得出一般拓扑中赫尔斯 - 乌拉姆型稳定性结果，还研究了特定拓扑情形及给出下限和三明治型结果。
英文摘要
The central question of this paper is the following: "If a topological space equipped with a weight function satisfies a certain property only approximately, can we recover the original structure without significantly altering the weights?'' To each open set of a topological space, we assign a non-negative weight and study approximate versions of three natural properties: monotonicity, subadditivity, and convexity. Through this study, we develop Hyers--Ulam-type stability results in general topology. In addition, we investigate several topology-specific cases, as well as present minorant and sandwich-type results.
