When a totally-ordered set Ω has no points fixed by all automorphisms ve equip Ω with the support topology which has fixed-sets of automorphisms as basic closed sets. There is a mapping from Ω into the set of prime dual ideals of the lattice Φ(Ω) of basic closed sets and this allows us to classify the points of Ω as excellent, isolated, static, or extraordinary. There is an action of the group of automorphisms of Ω on the lattice Φ(Ω) and this allows us to see that automorphisms of Ω preserve the prime dual ideal classification of points of Ω. When the empty set is a basic closed subset of Ω the dual spectrum of Φ(Ω) is a compact T0 space (Hausdorff when all points of Ω are excellent) containing Ω as a dense subspace and allowing an extension of each automorphism of Ω ťo a homeomorphism of the dual spectrum.