Concerning continuous images of compact ordered spaces

It is the purpose of this paper to prove that if each of X and Y is a compact Hausdorff space containing infinitely many points, and X X Y is the continuous image of a compact ordered space L, then both X and Y are metrizable.2 The preceding theorem is a generalization of a theorem [1 ] by Mardesi6 and Papi6, who assume that X, Y, and L are also connected. Young, in [3], shows that the Cartesian product of a "long" interval and a real interval is not the continuous image of any compact ordered space. In this paper, the word compact is used in the "finite cover" sense. The phrase "ordered space" means a totally ordered topological space with the order topology. A subset M of a topological space is said to be heriditarily separable provided each subset of M is separable. If a and b are points of an ordered space L and a<b, then [a, b] ((a, b)) will denote the set of all points x of L such that a < x ? b (a <x <b), provided there is one; also, [a, b] will be used even in the case where a = b. A subset M of an ordered space L is convex provided that if aEM, bEM, and a <b, then [a, b] CM. If M is a subset of an ordered space L, then G(M) will denote the set of all ordered pairs (a, b) such that (1) aEM, bEM, and a <b, and (2) {a, b }M [a,b], provided there is one.