Continuous structures in first-order Basic logic

The interpretation of logical connectives in continuous t-norm based fuzzy logics are not necessarily continuous. Recently two topologies are introduced on the sets [0,1] and [0,1]^2 where the interpretation of all logical connective under these topologies are continuous. In this paper, firstly we study some of the properties of these topologies and then besides the ideas of the first-order continuous logic, we introduce first-order continuous fuzzy logic based on a continuous t-norms and finally it is shown that the interpretation of any formula in this logic is a continuous function.