Cyclic Order - Topology

Topology

The open intervals form a base for a natural topology, the cyclic order topology. The open sets in this topology are exactly those sets which are open in every compatible linear order. To illustrate the difference, in the set [0, 1), the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order.

Interesting examples of cyclically ordered spaces include the conformal boundary of a simply connected Lorentz surface and the leaf space of a lifted essential lamination of certain 3-manifolds. Discrete dynamical systems on cyclically ordered spaces have also been studied.

The interval topology forgets the original orientation of the cyclic order. This orientation can be restored by enriching the intervals with their induced linear orders; then one has a set covered with an atlas of linear orders that are compatible where they overlap. In other words, a cyclically ordered set can be thought of as a locally linearly ordered space: an object like a manifold, but with order relations instead of coordinate charts. This viewpoint makes it easier to be precise about such concepts as covering maps. The generalization to a locally partially ordered space is studied in Roll (1993); see also Directed topology.