The Gruenhage property, property *, fragmentability, and $\sigma $-isolated networks in generalized ordered spaces

Harold Bennett ¹   ; David Lutzer ²

¹ Texas Tech University Lubbock, TX 79405, U.S.A.
² College of William and Mary Williamsburg, Va 23187, U.S.A.

Fundamenta Mathematicae, Tome 223 (2013) no. 3, pp. 273-294

Résumé

\looseness -16We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of $\sigma $-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 $\sigma $-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 $\sigma $-disjoint $\pi $-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 $\sigma $-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 $\sigma $-isolated network and a $\sigma $-disjoint $\pi $-base but contains no dense metrizable subspace.

Détail

DOI : 10.4064/fm223-3-4

Keywords: looseness examine gruenhage property property * introduced orihuela smith troyanski fragmentability existence sigma isolated networks context linearly ordered topological spaces lots generalized ordered spaces go spaces monotonically normal spaces monotonically normal space property * sigma isolated network hereditarily paracompact property * gruenhage property equivalent monotonically normal spaces however fragmentable monotonically normal space may fail paracompact fragmentable go space have sigma disjoint base follows theorem white fragmentable first countable go space has dense metrizable subspace go space fragmentable baire space has dense metrizable subspace compact lots metrizability equivalent each following eberlein compact talagrand compact gulko compact has sigma isolated network gruenhage space has property * perfect fragmentable function space * has strictly convex dual norm example go space has property * fragmentable has sigma isolated network sigma disjoint base contains dense metrizable subspace