Compact interval spaces in which all closed subsets are homeomorphic to clopen ones, II - To the memory of ernest corominas (1913-1992)

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space and a space is scattered if every non-empty subspace has an isolated point. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L^*, L+K, L · K denote respectively the reverse ordering of L, the ordered sum of L and K and the lexicographic order on L x K (so ω · 2=ω+ω). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals κ, γl ≥ 0, let L(K,λ)=K+1+λ*. Theorem: Let X be a compact interval scattered space. Then X is a CO space if and only if X is homeomorphic to a space of the form α+1+∑_1<nL(K_iλ_i), where α is any ordinal, n ∈ ω, for every i<n,K_i,λ_i are regular cardinals and K_i≥λ_i, and if n>0, then α≥max({K_i:i<n}). By Part I of this work, the hypothesis "scattered" is unnecessary.