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

To the memory of Ernest Corominas (1913–1992)

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  • 4 Citations

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 iλ 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).

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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

Communicated by E. C. Milner

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Bekkali, M., Bonnet, R. & Rubin, M. Compact interval spaces in which all closed subsets are homeomorphic to clopen ones, I. Order 9, 69–95 (1992). https://doi.org/10.1007/BF00419040

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Mathematics Subject Classifications (1991)

  • Primary 06B30
  • 54E45
  • 54E12. Secondary 06B05

Key words

  • Compact spaces
  • scattered spaces
  • order topology
  • Boolean algebras