uniform spaces

A uniform space is a mathematical framework that generalizes the concept of distance and closeness between points, beyond traditional metric spaces. It allows us to discuss how points can be considered "uniformly close" without assigning exact distances. This is achieved through a collection of "entourages"—special sets that specify which pairs of points are close to each other uniformly across the space. Uniform spaces are useful in analyzing concepts like continuity and convergence in more abstract contexts, providing a flexible foundation that encompasses metric spaces and other structures where the idea of uniform closeness is important.