Variable-basis topological systems versus variable-basis topological spaces (Q989684)

Topological systems were introduced by [\textit{S. Vickers}, Topology via logic. Cambridge Tracts in Theoretical Computer Science, 5. Cambridge etc.: Cambridge University Press. (1989; Zbl 0668.54001)] as a framework in which to treat both point-sensitive spaces (i.e. topological spaces), the respective algebraic structures underlying their topologies (i.e. frames) and the corresponding point-free spaces (i.e. locales). On the other hand, there is the category of variable-basis lattice-valued topological spaces introduced by [\textit{S. E. Rodabaugh}, Categorical foundation of variable-basis fuzzy topology. Mathematics of fuzzy sets. Logic, topology, and measure theory. Dordrecht: Kluwer Academic Publishers. Handb. Fuzzy Sets Ser. 3, 273--388 (1999; Zbl 0968.54003)]. In the present paper, a variable-basis generalization of topological systems over an arbitrary variety of algebras is introduced and the functorial relationships between the categories of variable-basis topological systems and variable-basis lattice-valued topological spaces are investigated. This allows to treat variable-basis lattice-valued topological spaces (in the sense of Rodabaugh) and the respective algebraic structures underlying their topologies in a single framework. Some intrinsic properties (specifically products and coproducts) of the category of variable-basis topological systems are considered.