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## Fundamenta Mathematicae

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

### Tom 223 / 2013

Fundamenta Mathematicae 223 (2013), 273-294 MSC: Primary 54F05; Secondary 54E52, 54D30, 46A99, 46B03, 46B26, 46B50. DOI: 10.4064/fm223-3-4

#### Streszczenie

\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.

#### Autorzy

• Harold BennettTexas Tech University
Lubbock, TX 79405, U.S.A.
e-mail
• David LutzerCollege of William and Mary
Williamsburg, Va 23187, U.S.A.
e-mail

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