When the property of having a $\pi$-tree is preserved by products

TL;DR

This paper establishes conditions under which the property of having a π-tree is preserved in product spaces, providing new examples and stability results for spaces with π-trees, including powers of the Sorgenfrey line.

Contribution

It introduces sufficient conditions for product spaces to retain π-trees and demonstrates this for specific spaces like the Sorgenfrey line and its powers.

Findings

At most countable powers of the Sorgenfrey line have a π-tree.

At most countable powers of the irrational Sorgenfrey line have a π-tree.

Product spaces with a π-tree retain this property when combined with Baire space, Sorgenfrey line, and its powers.

Abstract

We find sufficient conditions under which the product of spaces that have a $\pi$-tree also has a $\pi$-tree. These conditions give new examples of spaces with a $\pi$-tree: every at most countable power of the Sorgenfrey line and every at most countable power of the irrational Sorgenfrey line has a $\pi\!$-tree. Also we show that if a space has a $\pi$-tree, then its product with the Baire space, with the Sorgenfrey line, and with the countable power of the Sorgenfrey line also has a $\pi$-tree.