an:03964949
Zbl 0598.54003
Katsaras, A. K.
On fuzzy uniform spaces
EN
J. Math. Anal. Appl. 101, 97-113 (1984).
00150099
1984
j
54A40 54E15 54E05
fuzzy uniformity; ordinary uniform spaces; fuzzy uniform space; fuzzy topology; fuzzy proximity
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 \({\mathcal U}(\delta)\).