## 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).

Original language | English |
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Pages (from-to) | 69-95 |

Number of pages | 27 |

Journal | Order |

Volume | 9 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 1992 |

## Keywords

- Boolean algebras
- Compact spaces
- Mathematics Subject Classifications (1991): Primary 06B30, 54E45, 54E12. Secondary 06B05
- order topology
- scattered spaces

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics