Sums of L-fuzzy topological spaces

It follows from L-FTOP being topological over SET that there is L-fuzzy topological sum in L-FTOP. In this paper, we use the final topologies constructed in Section 3 of Rodabaugh [Powerset operator foundations of variable-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, the Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston/Dordrecht/London, 1999, pp. 91–116] to characterize L-fuzzy topological sum internally and establish connections between L-fuzzy topological sum and its factor spaces. We create a functor ω from L-FYS (the category of L-fuzzifying topological space) to L-FTOP and show that ω has a right-adjoint, hence there exists an adjunction from L-FYS to L-FTOP. Moreover ω preserves L-fuzzy topological sums that already exists in both L-FYS and L-FTOP. Finally, we examine certain additivity property of L-fuzzy topological spaces. These results imply that the topological sum in L-FTOP, a necessary consequence of L-FTOP being a topological construct, is now better understood.