In this paper, properties of connectedness on completely distributive topological lattices (Definition 2.1) and fuzzy topological spaces [1] are studied. Some equivalent propositions on the concepts of functions in completely distributive lattices, and connected elements in completely distributive lattices and fuzzy topological spaces are proved. Specially, connectedness is characterized both by the R-neighborhoods of molecules and C-separations. For any family {(L_i, τ_i): i ε I} of fuzzy topological spaces, the product [1] is connected iff every (L_i, τ_i) is connected. A continuous function maps connected elements onto connected ones. The connectedness on the fuzzy real line (L) is discussed in the last section as an important example. The intervals defined here (Definition 4.1) must be connected in the fuzzy real line (L), and so the intervals defined by Rodabaugh in [13] are too. Any connected element is connected in the fuzzy real line (L), if and only if (L) .