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

Abstract

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. 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 orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ω·2=ω+ω and 2·ω=ω). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals κ, λ≥0, let L(κ, λ)=κ + 1 + λ * . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form α + 1 + Σ i<n L(κ i , λ i ), where α is any ordinal, n∈ω, for every i<n, κi, λi are regular cardinals and κ i⩾λ i, and if n>0, then α⩾max({κ i: i<n}) · ω. This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).