If there exists no z in X such that x < z < y, consider the sets:A = (-∞, y)B = (x, +∞)These sets are disjoint (If a is in A, a < y so that if a is in B, a > x and a < y which is impossible by hypothesis), nonempty (x is in A and y is in B) and open (in the order topology) and their union is X.