On a topological model for the category of fuzzy topological spaces. II
Let j denote the hypergraph functor [see the review above]. \textit{A. J. Klein} [Rocky Mount. J. Math. 17, 15-18 (1987; Zbl 0621.54003)] proved that a fuzzy topological space (X,T) is topologically generated (i.e. \(T=\omega (\tau)\) where \(\omega\) (\(\tau\)) \(=\) all lower semicontinuous mappings from (X,\(\tau\)) to [0,1]) iff \(j(X,T)=(X\times [0,1)\), \(\tau \times \tau_{\ell})\) where \(\tau_{\ell}=\{[0,t):\) \(t\in [0,1]\}\). The author introduces the concept of an f-topologically generated fuzzy topological space: (X,T) is f-topological generated iff \(j(X,T)=(X\times [0,1),j(T))\) with \(j(T)=\tau \times \tau^*\), where \(\tau^*\subset \tau_{\ell}\). Several relationships between (X,T) and (X,\(\tau\)) are investigated. Unlike part I [see the review above], the author also considers properties which are not order-theoretic properties of the family of all fuzzy open sets (e.g. fuzzy compactness in the sense of \textit{R. Lowen} [J. Math. Anal. Appl. 56, 621-633 (1976; Zbl 0342.54003)]). Also, products of topological spaces which are images of fuzzy spaces under the hypergraph functor are investigated. \(\{\) Reviewer's remarks: The author only considers properties that are preserved under weakening of a fuzzy topology. Therefore, some of his results about f-topologically generated spaces are special cases rather than generalizations of the corresponding statements about topologically generated spaces. (In fact, they are equivalent but this is rather uninteresting equivalence.) For instance, Theorem 8.2 states that an f- topologically generated space (X,T) is fuzzy compact iff (X,\(\tau\)) is compact. However, one has \(T\subset \omega (\tau)\), hence the non-trivial part is immediate from the fact that \(\tau\) is compact iff \(\omega\) (\(\tau\)) is fuzzy compact (see Lowen [op. cit.])\(\}\).
scientific article; zbMATH DE number 4151342