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 (http://planetmath.org/OperationsOnRelations) of $U$ , and $\\circ$ is the composition of relations (http://planetmath.org/OperationsOnRelations). 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.

单词	GeneralizationOfAUniformity
释义	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 (http://planetmath.org/OperationsOnRelations) of $U$ , and $\\circ$ is the composition of relations (http://planetmath.org/OperationsOnRelations). 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.
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