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Fuzzy topology. (English) Zbl 0906.54006

The book is devoted to giving an account of fuzzy topology. There are 15 chapters which cover the following topics: fuzzy topological spaces, continuity, product spaces, quotient spaces, stratified spaces, induced spaces, convergence theory, connectedness, countability, L-valued weakly induced spaces, separation axioms, embedding theory, insertion theorem, N-compactness, the Stone-Čech compactification, paracompact spaces, uniformity, proximity, metrics in Hutton’s sense and Erceg’s sense, relations between fuzzy topological spaces and locales and fuzzy representation theorem.
Fuzzy topology is a kind of topology on a lattice, and so it involves many problems on ordered structure. Some proofs are based on the stratification method and the pointed approach in this book. To distinguish General Topology from Fuzzy Topology, it is imperative to quote some results from the book.
(1) N-compactness is a good L-extension of ordinary compactness.
(2) Stratified property is hereditary.
(3) Weakly induced property is strongly multiplicative. Here ‘multiplicative’ means ‘productive’.
(4) Let \((L^x,\delta)\) be a fuzzy topological space. The following conditions are equivalent: (a) \((L^x, \delta)\) is \(T_{1{1 \over 2}}\) and \(L\) is \(T_{1{1 \over 2}}\) as a locale. (b) \((L^x, \delta)\) is \(T_0\) and \(\delta\) is \(T_{1{1 \over 2}}\) as a locale.
(5) (Tikhonov Product Theorem) N-compactness of L-fuzzy topological spaces is strongly multiplicative.
The bibliography contains 201 references. Index is provided. The getup is attractive. Pre-requisites for reading this book are lattice theory, general topology and fuzzy set theory. The material is taken from research papers. As such that book is useful to teachers and researchers, who are interested in fuzzy topology. It is the first book on fuzzy topology. The reviewer congratulates the authors and the publisher for their maiden enterprise.

MSC:

54A40 Fuzzy topology
54-02 Research exposition (monographs, survey articles) pertaining to general topology
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