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2013 | 223 | 3 | 273-294
Tytuł artykułu

The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces

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EN
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EN
We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. We show that any monotonically normal space with property * or with a σ-isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces. (However, a fragmentable monotonically normal space may fail to be paracompact.) We show that any fragmentable GO-space must have a σ-disjoint π-base and it follows from a theorem of H. E. White that any fragmentable, first-countable GO-space has a dense metrizable subspace. We also show that any GO-space that is fragmentable and is a Baire space has a dense metrizable subspace. We show that in any compact LOTS X, metrizability is equivalent to each of the following: X is Eberlein compact; X is Talagrand compact; X is Gulko compact; X has a σ-isolated network; X is a Gruenhage space; X has property *; X is perfect and fragmentable; the function space C(X)* has a strictly convex dual norm. We give an example of a GO-space that has property *, is fragmentable, and has a σ-isolated network and a σ-disjoint π-base but contains no dense metrizable subspace.
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  • Texas Tech University, Lubbock, TX 79405, U.S.A.
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  • College of William and Mary, Williamsburg, Va 23187, U.S.A.
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-3-4
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