The Gruenhage property, property *, fragmentability, and \(\sigma \)-isolated networks in generalized ordered spaces (Q2866778)

The \textit{Gruenhage property} of a space \(X\) refers to the existence of a sequence \(\langle \mathcal{V}(n): n<\omega\rangle\) of collections of open sets such that given any points \(x\neq y\) in \(X\), there is some \(n\) and some \(V\in\mathcal{V}(n)\) such that \(V\cap\{x,y\}\) is a singleton, and at least one of the numbers \(\mathrm{ord}(x,\mathcal{V}(n))\) and \(\mathrm{ord}(y,\mathcal{V}(n))\) is finite. (The number \(\mathrm{ord}(x,\mathcal{V}(n))\) is the cardinality of the set \(\{V\in\mathcal{V}(n): x\in V\}\).) This property was introduced by \textit{G. Gruenhage} [Proc. Am. Math. Soc. 100, 371--376 (1987; Zbl 0622.54020)].NEWLINENEWLINEProperty * of a space (introduced by \textit{J. Orihuela} et al. in [Proc. Lond. Math. Soc. (3) 104, No. 1, 197--222 (2012; Zbl 1241.46005)]) is a generalization of the Gruenhage property. A space \(X\) is said to have \textit{property *} if there is a sequence \(\langle \mathcal{V}(n): n<\omega\rangle\) of collections of open sets such that if \(x\neq y\) are any two points of \(X\), there is some \(n\) and some \(V\in\mathcal{V}(n)\) such that \(V\cap\{x,y\}\) is a singleton and no member of \(\mathcal{V}(n)\) contains both \(x\) and \(y\).NEWLINENEWLINEThese properties, together with fragmentability (a space \(X\) is \textit{fragmentable} if there is a metric \(d\) on \(X\), not directly related to the topology \(\tau\) of \(X\), such that for each \(\varepsilon>0\) and each nonempty \(E\subseteq X\) there is some \(U\in\tau\) with \(U\cap E\neq \emptyset\) and the \(d\)-diameter of \(U\cap E\) is smaller than \(\varepsilon\)) and the existence of \(\sigma\)-isolated networks, are carefully investigated in this paper in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. To quote the abstract:NEWLINENEWLINE``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; and the function space \(C(X)^*\) has a strictly convex dual norm.''NEWLINENEWLINEAn 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 is presented at the end of the paper.