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

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.