Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is connected if any two elements that are related in neither order are equal:
This is a basic property of linear orders; an apartness relation is usually called tight if it is connected.
Using excluded middle, it is equivalent to say that every two elements are related in some order or equal:
However, this version is too strong for the intended applications to constructive mathematics. (In particular, $\lt$ on the located Dedekind real numbers satisfies the first definition but not this one.)
On the other hand, there is a stronger notion that may be used in constructive mathematics, if $A$ is already equipped with a tight apartness $\#$. In that case, we say that $\sim$ is strongly connected if any two distinct elements are related in one order or the other:
Since $\#$ is connected itself, every strongly connected relation is connected; the converse holds with excluded middle (through which every set has a unique tight apartness).