A representation theorem for order continuous Banach lattices

Suppose X is a topological space and S (X) is the vector lattice of all continuous functions on open dense subsets of X. Although, S (X) is not a normed lattice, we can have unbounded norm topology (un-topology) on it. On the other hand, the recent result, due to Wickstead, presents a representation approach for every Archimedean vector lattice E in terms of S (X) -spaces. In this note, we show that this representation is order continuous and when E is order complete, it coincides with the known Ogasawara-Maeda representation. Moreover, we obtain a similar representation for an order continuous Banach lattice in terms of S (X) -spaces that is a un-homeomorphism.