L-topological spaces

By substituting the usual notion of open sets in a topological space X with a suitable collection of maps from X to a frame L, we introduce the notion of L-topological spaces. Then, we proceed to study the classical notions and properties of usual topological spaces to the newly defined mathematical notion. Our emphasis would be concentrated on the well understood classical connectedness, quotient and compactness notions, where we prove the Thychonoff's theorem and connectedness property for ultra product of L-compact and L-connected topological spaces, respectively.