Approximate monotonicity, subadditivity, and convexity in weighted topologies

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.