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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
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