[Paper Review] Various topologies on trees

Peter Nyikos|ArXiv.org|Dec 31, 2004

Advanced Topology and Set TheoryMathematics12 references20 citations

TL;DR

This paper surveys multiple topologies on trees, focusing on the coarse and fine wedge topologies and the interval topology, demonstrating that the coarse wedge topology yields supercompact, monotone normal spaces, while the fine wedge topology induces a hereditarily ultraparacompact, monotone normal topology on every tree. The interval topology reveals deep set-theoretic dependencies, with properties like normality and metrizability often contingent on axioms beyond ZFC.

ABSTRACT

This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each element are well-ordered. A number of different topologies on trees are treated, some at considerable length. Two sections deal in some depth with the coarse and fine wedge topologies, and the interval topology, respectively. The coarse wedge topology gives a class of supercompact monotone normal topological spaces, and the fine wedge topology puts a monotone normal, hereditarily ultraparacompact topology on every tree. The interval topology gives a large variety of topological properties, some of which depend upon set-theoretic axioms beyond ZFC. Many of the open problems in this area are given in the last section.

Motivation & Objective

• To systematize and analyze various topological structures on trees, particularly wedge and interval topologies.
• To investigate the topological properties—such as monotone normality, ultraparacompactness, and metrizability—induced by different topologies on trees.
• To explore the dependence of topological properties on set-theoretic axioms beyond ZFC, especially in the context of Aronszajn and special trees.
• To resolve open problems concerning normality, metrizability, and paracompactness in tree topologies under different set-theoretic assumptions.
• To clarify the role of special and R-special trees in determining topological behavior, particularly in relation to normality and metrizability.

Proposed method

• Defines trees as partially ordered sets where predecessors of any element are well-ordered, establishing foundational structure.
• Introduces the coarse wedge topology, showing it induces supercompact, monotone normal spaces on all trees.
• Introduces the fine wedge topology, proving it yields a hereditarily ultraparacompact, monotone normal topology on every tree.
• Analyzes the interval topology, revealing its sensitivity to set-theoretic axioms such as CH, MA, and the existence of weak Kurepa trees.
• Applies the Pressing-Down Lemma and properties of stationary and nonstationary sets to characterize cwH (completely weakly hereditary) trees.
• Uses forcing extensions and large cardinal assumptions (e.g., MA + ¬wKH) to analyze consistency results for normality and metrizability.

Experimental results

Research questions

• RQ1 Under which set-theoretic assumptions is every normal tree monotonically normal?
• RQ2 Is it consistent that every perfectly normal tree is metrizable, independent of ZFC?
• RQ3 Can there exist a countably paracompact, non-normal Aronszajn tree under V = L?
• RQ4 Is every R-special, countably paracompact tree necessarily collectionwise normal?
• RQ5 What is the consistency strength of the existence of a non-special, countably metacompact tree with no uncountable branches?

Key findings

• The coarse wedge topology endows every tree with a supercompact, monotone normal topology.
• The fine wedge topology gives every tree a hereditarily ultraparacompact, monotone normal topology.
• Under MA + ¬wKH, every cwH tree of height less than ω₂ is monotonically normal.
• Every R-special, cwH, countably paracompact tree is (collectionwise) normal, as shown in Theorem 4.39.
• In models with MA + ¬CH and strongly compactly many random reals, PMEA holds and every normal, first-countable tree is cwn, hence metrizable.
• The existence of a σ-orderable tree is equivalent to being σ-orderable and cwH, as established in Theorem 4.45.