A Representation for Topologically Separable Chains Original Research Article

A characterization of chains which are separable and compact in their order topology is given in (A.J. Ostaszewski, J. London Math. Soc.7 (1974), 758-760) as those which can be order-embedded into two horizontal segments of the lexicographic square so that the image meets the lower segment in a closed subset and the upper in points whose abscissas occur among those of the image in the lower (what this comes to is parallel projection of the upper on the lower segment sending the image into itself). The proof there is prepared for by a preliminary two-page Section which introduces two equivalence relations and develops their properties in a sequence of four lemmas. A direct proof of a more general result, which also isolates the characteristics determining such a representation, is presented.