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

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, metrizabil- ity 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 prop- erty : 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.