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On fuzzy uniform spaces. (English) Zbl 0598.54003
We give some results about fuzzy uniform spaces. Our definition of a fuzzy uniform space is that of Hutton with the only difference that every member \(\alpha\) of a fuzzy uniformity in our sense is such that \(\alpha (0)=0\). This is in accordance with what happens in the ordinary uniform spaces. The notion of a fuzzy uniform space given by Lowen differs from our concept of a fuzzy uniform space. We show that to every uniformity \({\mathcal U}\) on a set X corresponds a fuzzy uniformity \(\phi\) (\({\mathcal U})\) and that to every fuzzy uniformity \(\Phi\) on X corresponds a uniformity \(\psi\) (\(\Phi)\). The fuzzy topology generated by a uniformizable topology is uniformizable. In the last section we prove that for every fuzzy proximity \(\delta\), the class \(\Pi\) (\(\delta)\) of all fuzzy uniformities which are compatible with \(\delta\) is not empty and that \(\Pi\) (\(\delta)\) contans a smallest member \({\mathcal U}(\delta)\).

MSC:
54A40 Fuzzy topology
54E15 Uniform structures and generalizations
54E05 Proximity structures and generalizations
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