The Dual of a Circle Order Is Not Necessarily a Circle Order

A partially ordered set \(P\) is called a circle order if one can assign to each element \(a \in P\) a circular disk in the plane \({C_a}\), so that \(a < b\) iff \(C_a \subset C_b\). It is known that the dual of every finite circle order is a circle order. We show that this is false for infinite circle orders.