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Fuzzy proximities and totally bounded fuzzy uniformities. (English) Zbl 0558.54002
As pointed out by the authors, the structures of fuzzy proximity introduced by {\it A. K. Katsaras} (ibid. 68, 100-110 (1979; Zbl 0412.54006) and 75, 571-583 (1980; Zbl 0443.54006)] are inadequate. Instead of the intersection by complementation (precisely, by quasi- coincidence ({\it Pu Paoming} and the reviewer, ibid. 76, 571-599 (1980; Zbl 0447.54006)], the authors succeed to establish a reasonable structure of fuzzy proximity. Some well-known classical theorems are nicely extended to fuzzy set theory. Among others, the authors show that every fuzzy uniformity [in the sense of {\it B. Hutton}, ibid. 58, 559-571 (1977; Zbl 0358.54008)] induces a fuzzy proximity and vice versa. Moreover, they prove that there exists a 1-1 correspondence between fuzzy proximity spaces and a subclass of fuzzy uniform spaces and that the correspondence is functorial. The subclass of fuzzy uniform spaces is called totally bounded. Some interesting results about the subclass are also given.
Reviewer: Liu Yingming

MSC:
54A40Fuzzy topology
54E05Proximity structures and generalizations
54E15Uniform structures and generalizations
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References:
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