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 $V'=\{(y,x)\mid (x,y)\in V\}$ 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 $x\sim 'y,y\sim 'z\implies x\sim z$.