Some aspects of fuzzy topology (Q1300187)

This is a survey of some achievements in fuzzy topology done by mathematicians (mainly) of Chinese school in the last twenty years. In particular, the following aspects are discussed: Multiple choice principle and its role in fuzzy topology, particularly, for the description of the neighborhood structure of an \(L\)-fuzzy space. By the multiple choice principle the author means the existence of a relation \(\Delta\) such that a fuzzy point is in relation \(\Delta\) with a join of a family of fuzzy sets iff it is in \(\Delta\) with at least one of these fuzzy sets. (As different from the belongness relation \(\leq\), the quasi-coincidence relation introduced by Liu and later extended by Wang for more general lattices \(L\) satisfies this principle.) The fuzzy unit interval, fuzzy cubes and embedding theorems. In particular, the author mentions results by Zhang Dexue on uniformization of a stratified \(L\)-fuzzy unit interval and theorems about embedding of \(L\)-fuzzy topological spaces satisfying certain separation properties into products of fuzzy unit intervals (Liu, Wang Guojun, Xu). Properties of compactness, paracompactness and construction of compactifications. Among others, results by Liu and Luo, Peng and Zhao are discussed here. Fuzzy metric spaces and metrization problem. As one of the most important results the fuzzy version of Bing-Nagata-Smirnov proved by Liang is mentioned here. Function spaces in fuzzy topology. A fuzzy version of the Ascoli theorem proved by Peng is mentioned. Results around order structure of the lattice \(L\). Some relations between the order on \(L\), and certain properties of \(L\)-fuzzy sets are studied. In particular, Liu and Liang solved Lawson's problem concerning Scott topology on a lattice \(L\). The role of the stratification axiom, in particular, in connection with the Hahn-Dieudonné-Tong-type insertion theorem. Some results in this direction obtained by Liu and Luo are discussed. Relations between fuzzy topology and theory of locales were studied in a series of papers by Liu and Zhang Dexue.