Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence.
According to Wikipedia, there are five equivalent definitions of a uniform space.
My question is: What is the concept of uniform space useful for? What is the intuition for that?
And what is a "uniformity of a function"? (One of my professors mentioned it.) Does it have to do anything with uniform spaces? (The context was something like describing some spaces using "uniformity of a function", I dont remember exactly.)
Thank you!