Axiomatic foundations of fixed-basis fuzzy topology (Q2702343)

This paper is a chapter in a handbook which attempts to give a comprehensive overview of the state of the art of fuzzy topology and related matters in fuzzy set theory.NEWLINENEWLINENEWLINEIn 1965, Zadeh introduced the idea of fuzzy sets of a set \(X\), which is defined to be a function \(X\to [0,1]\). In 1968, C. L. Chang combined the idea of fuzzy sets with topological ideas and thus introduced the concept of fuzzy topological space. A Chang fuzzy topology on a set \(X\) is a subframe of \([0,1]^X\), i.e., a subset of \([0,1]^X\) closed with respect to finite infs and arbitrary sups. In 1970, Goguen introduced the idea of \(L\)-fuzzy topological space replacing \([0,1]\) by a complete lattice \(L\).NEWLINENEWLINENEWLINEIn the 1980's, Šostak introduced a different idea of fuzzy topology which will be called Šostak topology at this moment. A Šostak topology on a set \(X\), by definition, is a function \(T:[0,1]^X\to[0,1]\) such that (1) \(T(0_X)= T (1_X)=1\); (2) \(T(\lambda_1\wedge \lambda_2)\geq T(\lambda_1)\wedge T(\lambda_2)\) and (3) \(T(\bigvee_{t\in T}\lambda_t)\geq \bigwedge_{t\in T}T(\lambda_t)\). Clearly, the unit interval \([0,1]\) in the definition of the Šostak topology can be replaced by a complete lattice \(L\).NEWLINENEWLINENEWLINEIn order to distinguish these two different ideas, it is agreed, in the present `Handbook' to call the Chang-Goguen spaces \(L\)-topological spaces and the Šostak topological spaces \(L\)-fuzzy topological spaces. Further, if the complete lattice \(L\) is enriched with another binary operation \(\otimes\) satisfying certain properties, then one can replace the meet operation \(\wedge\) by \(\otimes\) in the definitions of \(L\)-topological spaces and \(L\)-fuzzy topological spaces.NEWLINENEWLINENEWLINEFor a fixed lattice \(L\), one obtains a category of \(L\)-topological spaces and a category of \(L\)-fuzzy topological spaces. Clearly, for different lattices \(L\), one obtains different categories of \(L\)-topological spaces and \(L\)-fuzzy topological spaces. The complete lattices \(L\) are called the basis of the corresponding \(L\)-topological spaces \((L\)-fuzzy topological spaces).NEWLINENEWLINENEWLINEThis chapter is devoted to the categorical foundations of fixed-basis fuzzy topology, i.e., for the category of \(L\)-topological spaces \((L\)-fuzzy topological spaces) where \(L\) is a fixed lattice. Unlike most of the works in fuzzy topology, this chapter emphasizes that the meet operation \(\wedge\) in the postulations of \(L\)-topological spaces and \(L\)-fuzzy topological spaces could (and should) be replaced by an arbitrary binary operation \(\otimes\) on \(L\) which distributes over suprema. A systematical theory has been developed for such spaces, for example, the theory of convergence, the principle of \(L\)-continuous extension, etc. Particular attention is paid to the case when \(L\) is an \(MV\)-algebra and to the connection of fuzzy topology with other fields of mathematics.NEWLINENEWLINENEWLINETo summarize, this is a very interesting and substantial chapter which should not be overlooked by anyone working in fuzzy topology.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].