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

### Volume 223 / 2013

#### Abstract

\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.