I had the following curiosity/question related to the Suslin Problem. It would be very helpful if you could recommend to me some good literature on the topic. Thanks in Advance!
Suppose you have a non-empty set $(A, *)$ where $*$ is a binary relation that is transitive and antisymmetric. Additionally, suppose that:
(1) $(A, *)$ does not have a least nor a greatest element;
(2) $*$ is dense on $A$;
(3) $*$ is complete on $A$, in the sense that every non-empty bounded subset has a supremum and an infimum; and
(4) every collection of mutually disjoint non-empty open intervals in $(A, *)$ is countable
(Question:) Is $(A, *)$ necessarily order-isomorphic to the real line R?