The power of trees

Ari Meir Brodsky; Assaf Rinot; Shira Yadai

doi:10.1016/j.apal.2026.103776

The power of trees

Ari Meir Brodsky
, Assaf Rinot
, Shira Yadai

Department of Mathematics

Shamoon College of Engineering
Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

We give two consistent constructions of trees T whose finite power Tⁿ⁺¹ is sharply different from Tⁿ: • An ℵ₁-tree T whose interval topology X_T is perfectly normal, but (X_T)² is not even countably metacompact. • For an inaccessible κ and a positive integer n, a κ-tree such that all of its n-derived trees are Souslin and all of its (n+1)-derived trees are special.

Original language	English
Article number	103776
Journal	Annals of Pure and Applied Logic
Volume	177
Issue number	9
DOIs	https://doi.org/10.1016/j.apal.2026.103776
State	Published - 1 Oct 2026

Bibliographical note

Publisher Copyright:
© 2026 Elsevier B.V.

Keywords

Almost-Souslin trees
Countably metacompact spaces
Parameterized proxy principle
Perfectly normal spaces
Special trees
Square of a tree

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10.1016/j.apal.2026.103776

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abstract = "We give two consistent constructions of trees T whose finite power Tn+1 is sharply different from Tn: • An ℵ1-tree T whose interval topology XT is perfectly normal, but (XT)2 is not even countably metacompact. • For an inaccessible κ and a positive integer n, a κ-tree such that all of its n-derived trees are Souslin and all of its (n+1)-derived trees are special.",

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AU - Rinot, Assaf

AU - Yadai, Shira

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N2 - We give two consistent constructions of trees T whose finite power Tn+1 is sharply different from Tn: • An ℵ1-tree T whose interval topology XT is perfectly normal, but (XT)2 is not even countably metacompact. • For an inaccessible κ and a positive integer n, a κ-tree such that all of its n-derived trees are Souslin and all of its (n+1)-derived trees are special.

AB - We give two consistent constructions of trees T whose finite power Tn+1 is sharply different from Tn: • An ℵ1-tree T whose interval topology XT is perfectly normal, but (XT)2 is not even countably metacompact. • For an inaccessible κ and a positive integer n, a κ-tree such that all of its n-derived trees are Souslin and all of its (n+1)-derived trees are special.

KW - Almost-Souslin trees

KW - Countably metacompact spaces

KW - Parameterized proxy principle

KW - Perfectly normal spaces

KW - Special trees

KW - Square of a tree

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JO - Annals of Pure and Applied Logic

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IS - 9

M1 - 103776

ER -

The power of trees

Abstract

Bibliographical note

Keywords

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