Let $X$ be a set. Let $\mathcal{U}$ be a family of subsets of $X\times X$ such that $\mathcal{U}$ is a filter, and that every element of $\mathcal{U}$ contains the diagonal relation $\Delta$ (reflexive). Consider the following possible ``axioms'':

1. for every $U\in \mathcal{U}$ $U^{-1}\in \mathcal{U}$
2. for every $U\in \mathcal{U}$ there is $V\in \mathcal{U}$ such that $V\circ V\in U$

where $U^{-1}$ is defined as the inverse relation of $U$ and $\circ$ is the composition of relations. If $\mathcal{U}$ satisfies Axiom 1, then $\mathcal{U}$ is called a semi-uniformity. If $\mathcal{U}$ satisfies Axiom 2, then $\mathcal{U}$ is called a quasi-uniformity. The underlying set $X$ equipped with $\mathcal{U}$ is called a semi-uniform space or a quasi-uniform space according to whether $\mathcal{U}$ is a semi-uniformity or a quasi-uniformity.

A semi-pseudometric space is a semi-uniform space. A quasi-pseudometric space is a quasi-uniform space.

A uniformity is one that satisfies both axioms, which is equivalent to saying that it is both a semi-uniformity and a quasi-uniformity.

Bibliography

1

W. Page, Topological Uniform Structures, Wiley, New York 1978.