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A concept of completeness of quasi-uniform spaces. (English) Zbl 0723.54030

The author proposes a new definition of completeness for quasi-uniform spaces. He defines a quasi-uniform space (X,\({\mathcal U})\) to be quiet provided that it satisfies the following: For each \(U\in {\mathcal U}\) there is an entourage \(V\in {\mathcal U}\) such that if \(\{x'_{\alpha}|\alpha\in A\}\) and \(\{x''_{\beta},|\beta\in B\}\) are sets on X with \((x''_{\beta},x'_{\alpha})\to 0\) and \(x'\), \(x''\) are points of X such that \((x',x'_{\alpha})\in V\) for each \(\alpha\in A\) and \((x''_{\beta},x'')\in V\) for each \(\beta\in B\), then \((x',x'')\in U\). He shows that his theory of completeness is satisfactory for the class of quiet quasi-uniform spaces.

MSC:

54E15 Uniform structures and generalizations
54E52 Baire category, Baire spaces
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References:

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