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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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##### References:
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