Characterization of \(L\)-topologies by \(L\)-valued neighborhoods (Q2702345)

The chapter under review provides a characterization of stratified and transitive \(L\)-topologies in terms of crisp systems of \(L\)-valued neighborhood systems (these systems originate from [the author, Manuscripta Math. 38, 289-323 (1982; Zbl 1004.54500)]). A general assumption about the underlying lattice \(L\) is that \((L,\leq,*)\) is a complete MV-algebra. This assumption is sufficient to characterize probabilistic \(L\)-topologies in terms of those systems (an \(L\)-topology \(\tau\subset L^X\) is called probabilistic if \(\alpha\to g\in\tau\) whenever \(\alpha\in L\) and \(g\in\tau\), where \(\alpha\to \beta= \bigvee\{\gamma\in L: \alpha*\gamma\leq \beta\}\)). If \(L\) is completely distributive, then every stratified and transitive \(L\)-topology is proved to have a characterization by crisp systems of \(L\)-valued neighborhoods. For \(L= [0,1]\) (the real unit interval) it is shown that there exists the Booleanization of an \([0,1]\)-topology induced by a fuzzy neighborhood space in the sense of \textit{R. Lowen} [Fuzzy Sets Syst. 7, 165-189 (1982; Zbl 0487.54008)]. As a result of this, fuzzy neighborhood spaces have been characterized by two different types of many valued neighborhood spaces: by Boolean valued neighborhoods and by \([0,1]\)-valued neighborhoods. It is also shown that \([0,1]\)-fuzzifying topologies (as defined by \textit{M. Ying} [Fuzzy Sets Syst. 39, 303-321 (1991; Zbl 0718.54017)] and fuzzy neighborhood spaces are equivalent concepts.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].