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A theory of fuzzy uniformities with applications to the fuzzy real lines. (English) Zbl 0637.54007
Summary: Let L be a completely distributive lattice with order-reversing involution. The fuzzy real line $${\mathbb{R}}(L)$$ is uniformizable by a uniformity which both generates the canonical (fuzzy) topology and induces a pseudometric generating the canonical topology. If L is also a chain,, the usual addition and multiplication defined on $${\mathbb{R}}\equiv {\mathbb{R}}(\{0,1\})$$ extend jointly (fuzzy) continuously to $$\oplus$$ and $$\otimes$$ on $${\mathbb{R}}(L)$$. Three fundamental questions in fuzzy sets until now are the following: A. If $$L_ 1\simeq L_ 2$$, is $${\mathbb{R}}(L_ 1)$$ uniformly isomorphic to $${\mathbb{R}}(L_ 2)$$ in some sense? B. For each chain L, is $$\oplus$$ (jointly) uniformly continuous in a sense which guarantees its (joint) continuity on $${\mathbb{R}}(L)?$$ C. Is $${\mathbb{R}}(L)$$ a complete pseudometric space in some sense?
Categories $${\mathbb{Q}}{\mathbb{U}}$$ and $${\mathbb{U}}$$ are constructed using the [quasi-] uniformities of B. Hutton which enable us to answer these questions in the affirmative. These results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative: Does fuzzy topology have deep, specific, canonical examples?

##### MSC:
 54A40 Fuzzy topology 54E15 Uniform structures and generalizations 54B30 Categorical methods in general topology
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