Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,II |
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Authors: | Mohamed Bekkali Robert Bonnet Matatyahu Rubin |
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Affiliation: | (1) University of Colorado, Boulder, USA;(2) Département de Mathématiques, Université d'Aix-Marseille, 13397 Marseille Cedex 13, France;(3) Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel |
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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 · 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+ 1L(Ki i), where is any ordinal, n , for every ii, i are regular cardinals and Ki![ges](/content/rgk3860636774q00/xxlarge10878.gif) i, and if n>0, then ![agr](/content/rgk3860636774q00/xxlarge945.gif) max({Ki: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. |
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Keywords: | Primary 06B30 54E45 54E12 Secondary 06B05 |
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