Decomposing the real line into everywhere isomorphic suborders

Abstract

We show that if R = A ∪ B \mathbb {R} = A \cup B is a partition of R \mathbb {R} into two suborders A A and B B , then there is an open interval I I such that A ∩ I A \cap I is not order-isomorphic to B ∩ I B \cap I . The proof depends on the completeness of R \mathbb {R} , and we show in contrast that there is a partition of the irrationals R ∖ Q = A ∪ B \mathbb {R} \setminus \mathbb {Q} = A \cup B such that A ∩ I A \cap I is isomorphic to B ∩ I B \cap I for every open interval I I . We do not know if there is a partition of R \mathbb {R} into three suborders that are isomorphic in every open interval.