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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.

##### MSC:
 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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