A Classification of Ordinal Topologies

Ordinals carry a natural topology induced by their linear order. In this note, we classify the homeomorphism types of all ordinal topologies using the Cantor normal form and the notion of the Cantor–Bendixson rank of a point. Let < be a linear order on a set X. The order topology on X is generated by the subbase of rays of the form leftX(b) := {x ∈ X : x < b} and rightX(a) := {x ∈ X : a < x} for a, b ∈ X. Natural examples of order topologies are the standard topologies on N, Q, and R. Ordinals provide another source for natural order topologies; we call these ordinal topologies. These topologies have many discrete points: if ξ < ξ + 1 < α, then ξ + 1 is a discrete point and the topological space α splits disjointly into the open sets leftα(ξ + 1) and rightα(ξ). We shall provide a complete homeomorphic invariant for ordinal topologies, using the Cantor normal form and the notion of the Cantor-Bendixson rank. The intuition behind this classification is that ordinal topologies are mainly determined by their limit points. Recall that every nonzero ordinal α can be written uniquely in Cantor normal form (to base ω) as α = ω0 · k0 + · · · + ω αn · kn, where α ≥ α0 > · · · > αn and 0 < ki < ω for 0 ≤ i ≤ n [Je03, Theorem 2.26]. We define the limit complexity of α as lc(α) := α0, the coefficient of α as c(α) := k0 and the purity of α as p(α) := { 0 if α = ω · c(α) ≥ ω, and ωn otherwise. We call α a pure limit ordinal if p(α) = 0, and an impure limit ordinal if p(α) > 1. Of course, α is a successor ordinal iff p(α) = 1. We shall show that these three data provide a complete homeomorphic invariant for ordinal topologies: Main Theorem. For any two ordinals α and β, the ordinal topologies on α and β are homeomorphic (written α ∼= β) if and only if 〈lc(α), c(α),p(α)〉 = 〈lc(β), c(β),p(β)〉. 2000 Mathematics Subject Classification. Primary 03E10; 06F30.