TITLE: Characterising subsets of the reals as ordered spaces
QUESTION [6 upvotes]: There are concise and elegant characterisations of the real line as a topological space and as an ordered space in the literature.  I am interested in the harder case of characterising subsets of the reals in this manner.  There are satisfactory answers to the topological version (e.g., de Groot, Mary Ellen Rudin) which are, as to be expected, more complicated and inticrate in proof than for the whole space.  I recall reading a solution for the corresponding result in the category of ordered spaces but the standard search methods have failed to locate it.  Can anybody on this site assist me with a reference?

REPLY [4 votes]: Just for the record, the result seems to be due to Isodore Fleischer ("Numerical representation of utility", Jour. Soc. Ind. Appl. Math, 9 (1961) 48-50).  As the title indicates, it is useful in  determining when a suitable ordering on a state space is induced by a numerical function (price, utility function, temperature, entropy) and  unifies many such results in fields such as economics, thermodynamics, philosophy ...  This (surprisingly late) reference is the earliest one that I can trace.