General Topology/Uniform spaces

In this definition, if $(x,y)$ are contained in a sufficiently small entourage, they are considered "close" to each other. That is, a uniform structure provides a means of determining when two arbitrary points $x,y\in X$ are close. This is the intuition behind this definition. A very important special case of a uniform space are metric spaces, which we'll learn about in the next chapter. Uniform spaces are a generalisation of metric spaces, and many of the notions and theorems carry over from metric spaces to uniform spaces, and we'll immediately treat them in full generality.

A uniform structure induces a topology on its space.

Henceforth, we shall consider a uniform space as a topological space with this topology.

The following is a generalisation of the Heine–Borel theorem.

Exercises

1. Let $X$ be a set, and let $𝒰$ and $𝒱$ be uniform structures on $X$ so that they generate the same topology $\tau$ and $X$ is compact with respect to $\tau$. Prove that in fact $𝒰=𝒱$.
2. Let $X$ be a topological space whose topology is induced by both of the two uniform structures $𝒰$ and $𝒱$. Suppose that $X$ is complete with respect to the uniform structure induced by $𝒰$. Show that $X$ is complete with respect to the uniform structure induced by $𝒱$.