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Uniform spaces are topological spaces with additional structure which 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, you can formalize the idea that "x is as close to a as y is to b", while in a topological space you can only formalize "x is as close to a as y is to a".
Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach which is often useful in functional analysis.
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