Let X be a Banach lattice. A well-known problem arising from the theory of risk measures asks when order closedness of a convex set in X implies closedness with respect to the topology σ (X , X n ∼) , where X n ∼ is the order continuous dual of X. Motivated by the solution in the Orlicz space case, we introduce two relevant properties: the disjoint order continuity property (DOCP) and the order subsequence splitting property (OSSP). We show that when X is monotonically complete with OSSP and X n ∼ contains a strictly positive element, every order closed convex set in X is σ (X , X n ∼) -closed if and only if X has DOCP and either X or X n ∼ is order continuous. This in turn occurs if and only if either X or the norm dual X ∗ of X is order continuous. We also give a modular condition under which a Banach lattice has OSSP. In addition, we also give a characterization of X for which order closedness of a convex set in X is equivalent to closedness with respect to the topology σ (X , X uo ∼) , where X uo ∼ is the unbounded order continuous dual of X. [ABSTRACT FROM AUTHOR]