Lowen's compactness in \(L\)-fuzzy topological spaces
\textit{R. Lowen's} concept of compactness of a fuzzy topological space [J. Math. Anal. Appl. 56, 621-633 (1976; Zbl 0342.54003)] was extended by \textit{Wang Goujun} [Shanxi Normal Univ. Press, Xi'an, China, 1988 (in Chinese)] to the case of \(L\)-fuzzy topological spaces where \(L\) is a completely distributive complete bounded lattice with an order-reversing involution. In this extension \(\vee\)-irreducible non-zero elements of \(L\) (so-called molecules of \(L\)) play a crucial role. In this paper some new characterizations of Lowen's compactness of \(L\)- fuzzy topological spaces are obtained. Among them characterizations by families of remote neighbourhoods, by \(\alpha\)-filters and by families of \(L\)-fuzzy sets with finite \(\alpha\)-intersection property (here \(\alpha\) is a molecule in \(L\)). Besides it is proved that a topological space \((X,T)\) is compact iff the fuzzy topological space \((L^ X,\omega(T))\) is Lowen's compact, where \(\omega(T)\) denotes the family of all mappings \(U: (X,T) \to L\) such that all \(\{x: U(x) \leq a\}\), \(a\in L\), are closed in \((X,T)\).
scientific article; zbMATH DE number 487477