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 $$\mathcal U$$ and $$\mathcal V$$ 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 $$\mathcal U = \mathcal V$$.
 * 2) Let $$X$$ be a topological space whose topology is induced by both of the two uniform structures $$\mathcal U$$ and $$\mathcal V$$. Suppose that $$X$$ is complete with respect to the uniform structure induced by $$\mathcal U$$. Show that $$X$$ is complete with respect to the uniform structure induced by $$\mathcal V$$.