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# generalization of a uniformity

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.

# References

- 1
W. Page,
*Topological Uniform Structures*, Wiley, New York 1978.

## Mathematics Subject Classification

54E15*no label found*

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