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

Harold Bennett, and David Lutzer. "The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces." Fundamenta Mathematicae 223.3 (2013): 273-294. <http://eudml.org/doc/283316>.

@article{HaroldBennett2013,
abstract = {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.},
author = {Harold Bennett, David Lutzer},
journal = {Fundamenta Mathematicae},
keywords = {Gruenhage space; property *; fragmentable space; -isolated network; LOTS; linearly ordered topological space; GO-space; generalized ordered space; paracompactness; stationary sets; dense metrizable subspace; monotone normality; metrizability; -diagonal; -disjoint -base; quasi-developable space; Sorgenfrey line; Michael line; Eberlein compact; Talagrand compact; Gulko compact; strictly convex dual norm},
language = {eng},
number = {3},
pages = {273-294},
title = {The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces},
url = {http://eudml.org/doc/283316},
volume = {223},
year = {2013},
}

TY - JOUR
AU - Harold Bennett
AU - David Lutzer
TI - The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces
JO - Fundamenta Mathematicae
PY - 2013
VL - 223
IS - 3
SP - 273
EP - 294
AB - 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.
LA - eng
KW - Gruenhage space; property *; fragmentable space; -isolated network; LOTS; linearly ordered topological space; GO-space; generalized ordered space; paracompactness; stationary sets; dense metrizable subspace; monotone normality; metrizability; -diagonal; -disjoint -base; quasi-developable space; Sorgenfrey line; Michael line; Eberlein compact; Talagrand compact; Gulko compact; strictly convex dual norm
UR - http://eudml.org/doc/283316
ER -