Compact interval spaces in which all closed subsets are homeomorphic to clopen ones, II

Abstract

A topological spaceX whose topology is the order topology of some linear ordering onX, is called aninterval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called aCO space and a space isscattered if every non-empty subspace has an isolated point. We regard linear orderings as topological spaces, by equipping them with their order topology. IfL andK are linear orderings, thenL ^*, L+K, L · K denote respectively the reverse ordering ofL, the ordered sum ofL andK and the lexicographic order onL x K (so ω · 2=ω+ω). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals κ, γl ⩾ 0, letL(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<n L(K _i λ _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.