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,Ki,λi are regular cardinals and Ki⩾λi, and if n>0, then α⩾max({Ki:i<n}). By Part I of this work, the hypothesis “scattered” is unnecessary.
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Communicated by E. C. Milner
Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016.
Supported by the City of Lyon.
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Bekkali, M., Bonnet, R. & Rubin, M. Compact interval spaces in which all closed subsets are homeomorphic to clopen ones, II. Order 9, 177–200 (1992). https://doi.org/10.1007/BF00814409
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DOI: https://doi.org/10.1007/BF00814409