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

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 · 2=+). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , l 0, letL(K,)=K+1+*.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 +1+_{1 L(K i i), where is any ordinal, n , for every ii},_i are regular cardinals and K_ii, and if n>0, then max({K_i:iscattered is unnecessary.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.