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Metrizability, monotone normality, and other strong properties in trees. (English) Zbl 0969.54026
A tree is a partially ordered set in which the predecessors of any element are well-ordered. A chain in a tree is a totally ordered subset. An antichain in a tree is a set of pairwise incomparable elements. A tree is special if it is a countable union of antichains. The interval topology on a tree \(T\) is the topology with base all sets of the form \((s, t]= \{x\in T: s< x\leq t\}\), together with all singletons \(\{t\}\) where \(t\) is a minimal element of \(T\). Given a faithfully indexed family \(S= \{S_\alpha: \alpha\in A\}\) of disjoint subsets of a set \(X\), an expansion of \(S\) is a family \(\{U_\alpha:\alpha\in A\}\) such that \(U_\alpha\cap\bigcup S= S_\alpha\) for all \(\alpha\in A\). A space \(X\) is [strongly] collectionwise Hausdorff if every closed discrete subspace expands to a disjoint [respectively discrete] collection of open sets. A space \(X\) is monotone normal if to each pair \(\langle G,x\rangle\) where \(G\) is an open set and \(x\in G\), it is possible to assign an open set \(G_x\) such that \(x\in G_x\subset G\) so that \(G_x\cap H_y\neq\emptyset\) implies either \(x\in H\) or \(y\in G\). The following results are typical and deal with trees that are Hausdorff in the interval topology.
Theorem. Let \(T\) be a tree. The following are equivalent: (1) \(T\) is monotone normal and special; (2) \(T\) is strongly collectionwise Hausdorff and special; (3) \(T\) is collectionwise Hausdorff and special; (4) \(T\) is metrizable.
Theorem. Let \(T\) be a tree. The following are equivalent: (1) \(T\) is monotone normal; (2) \(T\) is the topological direct sum of totally ordered subspaces, each homeomorphic to an ordinal and each convex in \(T\).
Theorem. ZF is enough to imply that monotone normal special trees are metrizable.

54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54-02 Research exposition (monographs, survey articles) pertaining to general topology
54A35 Consistency and independence results in general topology
54E35 Metric spaces, metrizability
54D30 Compactness
54G20 Counterexamples in general topology
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