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generalization of a uniformity∗ CWoo† 2013-03-21 22:05:00 Let X be a set. Let U be a family of subsets of X × X such that U is a filter, and that every element of U contains the diagonal relation ∆ (reflexive). Consider the following possible “axioms”: 1. for every U ∈ U, U −1 ∈ U 2. for every U ∈ U, there is V ∈ U such that V ◦ V ∈ U , where U −1 is defined as the inverse relation of U , and ◦ is the composition of relations. If U satisfies Axiom 1, then U is called a semi-uniformity. If U satisfies Axiom 2, then U is called a quasi-uniformity. The underlying set X equipped with U is called a semi-uniform space or a quasi-uniform space according to whether 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. ∗ hGeneralizationOfAUniformityi created: h2013-03-21i by: hCWooi version: h38937i Privacy setting: h1i hDefinitioni h54E15i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1