Uniform space

This is a variation of topological space. View other variations of topological 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 $V^{\prime }=\{(y,x)\mid (x,y)\in V\}$ is also in $\mathcal{U}$.
5. If $V\in \mathcal{U}$, there exists a set $V^{\prime }$ such that whenever $(x,y)\in V^{\prime }$, $(y,z)\in V^{\prime }$, 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 ${\sim }^{\prime }\in \mathcal{U}$ such that $x{\sim }^{\prime }y,y{\sim }^{\prime }z⟹x\sim z$.