Categorical foundation of variable-basis fuzzy topology (Q2702344)

This paper is a companion chapter of the chapter by \textit{U. Höhle} and \textit{A. Šostak}, Axiomatic foundations of fixed-basis fuzzy topology, ibid., 123-272 (1999). In 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 Šostak topology can be replaced by a complete lattice \(L\).NEWLINENEWLINENEWLINEIn order to distinguish these two different ideas, the Linz Group (a group of mathematicians who participate the annual Seminar on fuzzy sets and related topics hold in Linz) agreed to call the Chang-Goguen spaces \(L\)-topological spaces and the Šostak topological spaces \(L\)-fuzzy topological spaces. More generally, U. Höhle observed that 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 (see the chapter by \textit{U. Höhle} and \textit{A. Šostak} [loc. cit.]). These terminologies are adopted in this paper.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). The above-mentioned chapter by U. Höhle and A. Šostak 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.NEWLINENEWLINENEWLINEThis paper provides a categorical foundation for \(L\)-topological spaces and \(L\)-fuzzy topological spaces when \(L\) is allowed to change from one to another object within a given category of complete lattices. It is proved that the category of \(L\)-topological spaces (\(L\)-fuzzy topological spaces), where \(L\) is allowed to change within a given category of complete lattices, is topological over some ground category (or base category); characterizations of some basic topological concepts such as continuity of functions, sobriety of spaces, etc., and numerous examples are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].