Branch space representations of lines
A tree is a partially ordered set \((T,\leq_T)\) such for each \(t\in T\) the set \(T_t=\{s\in T:s\neq t,s\leq_Tt\}\) is well ordered by \(\leq _T\). By a path in \(T\) we mean a subset \(\rho\) of \(T\) that is linearly ordered by \(\leq _T\) and has the property that if \(s\leq_Tt\) and \(t\in\rho\), then \(s\in\rho\). Each non-maximal path \(\rho\) determines a node of the tree by Node\((\rho)=\{t\in T:T_t=\rho\}\). By a branch of \(T\) we mean a maximal path in \(T\) and we denote the set of all branches of \(T\) by \(B_T\). For any \(b\in B_T\) let ht\((b)\) be the order type of the well-ordered set \(b\) (ordered as a subset of \((T,\leq_T) \)). For \(\alpha< \text{ht}(b)\) let \(b(\alpha)\) be the unique point of \(b\cap T_\alpha\). For distinct \(b,c\in B_T\), there is a first ordinal \(\delta=\Delta _T(b,c)\) such that \(b(\delta )\neq c(\delta)\). Then \(b(\delta),c(\delta)\) belong to the same node \(N\) of \(T\) and we define \(b<_Bc\) if and only if in the ordering \(<_N\) chosen for \(N\), we have \(b(\delta)<_Nc(\delta)\). The branch space of the tree is the linearly ordered set \((B_T,\leq_B)\). In this paper the authors investigate representing linearly ordered sets as branch spaces of trees satisfying certain natural restrictions designed to make them less complicated than the set are seeking to represent. The authors show that any uncountable order-complete linearly ordered set \(X\) can be represented as the branch space of a tree \(T\) that is more simple than \(X\) and that if \(X\) is representable as a branch space, then any \(G_\delta\)-subset of \(X\) is also representable as a branch space. However, they show that even though the usual set \(\mathbb{R}\) of real numbers can be represented as the branch space of a tree with countable height and countable levels, most subsets of \(\mathbb{R}\) cannot represented in this way. The authors characterize those limit ordinals \(\lambda\) that can be represented by trees whose nodes do not contain copies of \(\lambda\) or \(\lambda^*\) and study topological properties of branch spaces. These results show that if \(T\) is a tree of height \(\omega_1\) that does not contain any \(\omega_1\)-branches, then the branch space of \(T\) must be hereditarily paracompact. The authors investigate branch space of Aronszajn trees (trees with height \(\omega _{1}\) that have countable levels and countable branches), showing that any such branch space is hereditarily paracompact, first countable, Lindelöf, non-separable, non-metrizable, and provided \(T\) does not contain any Souslin subtree, then its branch space is not perfect (i.e., has a closed subset that is not a \(G_\delta\)-set). Also, the authors study the existence of \(\sigma\)-disjoint bases in branch spaces of trees.
scientific article; zbMATH DE number 2174915