Suslin's problem

In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set R with the four properties R does not have a least nor a greatest element; the order on R is dense (between any two distinct elements there is another); the order on R is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R), is R necessarily order-isomorphic to the real line R? If the requirement for the countable chain condition is replaced with the requirement that R contains a countable dense subset (i.e., R is a separable space), then the answer is indeed yes: any such set R is necessarily order-isomorphic to R (proved by Cantor).