© 1996, Topology Atlas

Let X be a set with topology script T. Define this space to be (linearly) orderable if there is a linear order on X such that the collection of open rays in X form a subbasis for script T.

The earliest work in this area was a topological characterization of certain subsets of the real line. The first orderability result that I am aware of was proved in 1905 by O. Veblen (Theory of plane curves in nonmetrical analysis situs, Trans. Amer. Math. Soc. 6 (1905), 83-98). He showed that every metric continuum with exactly two noncut points is homeomorphic to the unit interval. But the richest source of these early results is in volumes 1 and 2 of Fundamenta Mathematicae, 1920-21, containing articles by Mazurkiewicz, Sierpinski and Souslin. Beginning in 1941 characterizations of orderability for other classes were found, such as for connected spaces, metric spaces and scattered spaces.

Along with metric spaces, orderable spaces are the most basic spaces which satisfy strong separation axioms. Thus, it is natural to investigate whether some property holds for orderable spaces or their subspaces (suborderable spaces). (For reference see Topology and Order Structures, Parts I and II, H. R. Bennett and D. J. Lutzer (eds.), Math. Centre, Amsterdam, 1981 and 1983.) Some of these investigations led to other areas of mathematics. For example, properties of continuous images of orderable space are related to continuum theory, and order types and the duality between trees and linearly ordered sets led to consistency results in set theory.

An area of increasing activity in recent years is the relationship of orderable spaces to the larger class of monotonically normal spaces. For example, in 1973 I conjectured that monotone normality is equivalent to orderability for compact, separable, totally disconnected spaces. A counterexample is given in J. Nikiel, S. Purisch, and L. B. Treybig, Separable zero-dimensional spaces which are continuous images of ordered compacta, submitted. This leaves open the problem of characterizing when compact, separable, totally disconnected spaces are orderable. For further details on monotone normality see M. E. Rudin's problems on monotone normality, Topology Atlas Invited Contributions, vol. 1, issue 2 (1996), page 13.

J. van Dalen and E. Wattel characterized orderable spaces using a subbasis property in: A topological characterization of ordered spaces, Gen. Top. Appl. 3 (1973), 347-354. However, the subbase induces a unique order up to inverse on the space. So an open problem is to find a general characterization of orderable spaces not related to any particular linear order on a space.

Other open problems concerning properties of orderable spaces are Nyikos' question whether it is consistent that all perfect nonarchimedian spaces are metrizable and Gruenhage's question whether every uncountable regular space contains an uncountable suborderable subspace.

A survey article on properties of ordered spaces is D.J. Lutzer's Ordered topological spaces, in: George M. Reed (ed.), Surveys in General Topology, Academic Press, New York, 1980, 247-295.