Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence.

The conceptual difference between uniform and topological structures is that, in a uniform space, one can formalize certain notions of relative closeness and closeness of points. In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces. By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the closure of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.

Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis.

There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure.

This definition generalizes the presentation of a topological space in terms of neighborhood systems. A nonempty collection ${\displaystyle \Phi }$ of subsets ${\displaystyle U\subseteq X\times X}$ is a uniform structure if it satisfies the following axioms:

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