Some Results on Finitely Splitting Subtrees of Aronszajn Trees

TL;DR

This paper introduces a forcing method to add and analyze finitely splitting subtrees of Aronszajn trees, leading to new consistency results and solving an open problem about Suslin trees and their topological properties.

Contribution

It develops a new forcing poset using generalized promises to add normal n-splitting subtrees and proves several consistency results regarding splitting properties of Aronszajn trees.

Findings

Existence of an infinitely splitting Suslin tree with a non-Lindelöf topological square.

Consistency that every (≥ n)-splitting normal Aronszajn tree contains an n-splitting subtree.

Existence of a normal infinitely splitting Aronszajn tree with no (< n)-splitting subtree.

Abstract

For any $2 \le n < \omega$, we introduce a forcing poset using generalized promises which adds a normal $n$-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindel\"{o}f, which solves an open problem due to Marun. For any $2 < n < \omega$, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal $n$-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.