## 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, κ_{i}, λ_{i} 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*).

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## Additional information

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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### Cite this article

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