Adjacent pair in a linear order ($u<v\wedge \neg \exists w(u<w<v)$)

Two elements $u<v$ of a totally ordered set form an adjacent pair when no domain element lies strictly between them. Every pair of consecutive integers is adjacent, while a dense order has no adjacent pair. A gap in a subset of the real numbers matters here only through which points belong to the domain: $1,2$ are adjacent in $[0,1]\cup [2,3]$.