generalization of a uniformityGeneralizationOfAUniformity2007-02-20 13:52:472007-04-21 09:47:52Definitionuniform spacesemi-uniformityquasi-uniformitysemi-uniform spacequasi-uniform space\usepackage{amssymb,amscd}
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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'':
\begin{enumerate}
\item for every $U\in \mathcal{U}$, $U^{-1}\in \mathcal{U}$
\item for every $U\in \mathcal{U}$, there is $V\in \mathcal{U}$ such that $V\circ V\in U$,
\end{enumerate}
where $U^{-1}$ is defined as the \PMlinkname{inverse relation}{OperationsOnRelations} of $U$, and $\circ$ is the \PMlinkname{composition of relations}{OperationsOnRelations}. If $\mathcal{U}$ satisfies Axiom 1, then $\mathcal{U}$ is called a \emph{semi-uniformity}. If $\mathcal{U}$ satisfies Axiom 2, then $\mathcal{U}$ is called a \emph{quasi-uniformity}. The underlying set $X$ equipped with $\mathcal{U}$ is called a \emph{semi-uniform space} or a \emph{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.
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\bibitem{wp} W. Page, \emph{Topological Uniform Structures}, Wiley, New York 1978.
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