Suslin's problem

It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.

Suslin's problem asks: Given a non-empty totally ordered set R with the four properties is R necessarily order-isomorphic to the real line R?

The condition for a topological space that every collection of non-empty disjoint open sets is at most countable is called the Suslin property.

Jech (1967) and Tennenbaum (1968) independently used forcing methods to construct models of ZFC in which Suslin lines exist.

Jensen later proved that Suslin lines exist if the diamond principle, a consequence of the axiom of constructibility V = L, is assumed.