Uniform space

Definition
A uniform space is a set equipped with an additional structure called a uniform structure. A uniform structure on a set $$X$$ is a collection $$\mathcal{U}$$ of subsets of $$X \times X$$ (called entourages or vicinities) satisfying the following:

(In the language of sets):


 * 1) If $$A \subseteq B$$ and $$A \in \mathcal{U}$$, then $$B \in \mathcal{U}$$.
 * 2) A finite intersection of member of $$\mathcal{U}$$ is again in $$\mathcal{U}$$.
 * 3) Every member of $$\mathcal{U}$$ contains the diagonal.
 * 4) If $$V \in \mathcal{U}$$, the set $$V' = \{ (y,x) \mid (x,y) \in V \}$$ is also in $$\mathcal{U}$$.
 * 5) If $$V \in \mathcal{U}$$, there exists a set $$V'$$ such that whenever $$(x,y) \in V'$$, $$(y,z) \in V'$$, we have $$(x,z) \in V$$.

(In the language of relations): Here, we think of $$\mathcal{U}$$ as a collection of binary relations on $$X$$:


 * 1) If a relation is in $$\mathcal{U}$$, so is every coarser relation.
 * 2) The conjunction of a finite number of relations in $$\mathcal{U}$$ is also in $$\mathcal{U}$$.
 * 3) Every relation in $$\mathcal{U}$$ is reflexive.
 * 4) For any relation in $$\mathcal{U}$$, the mirror-image relation (where $$x$$ is related to $$y$$ iff $$y \sim x$$) is also in $$\mathcal{U}$$.
 * 5) If $$\sim \in\mathcal{U}$$, there exists a relation $$\sim' \in \mathcal{U}$$ such that $$x \sim' y, y \sim' z \implies x \sim z$$.