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

Abstract

In this paper we examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of -isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and mono- tonically normal spaces. We show that any monotonically normal space with property * or with a -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 -disjoint -base and it follows from a theorem of H.E. White that any fragmentable, rst-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 -isolated network; X is a Gruenhage space; X has property *; X is perfect and fragmentable; and 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 -isolated network and a -disjoint -base but contains no dense metrizable subspace.