The study of the interval topology of trees traces back to the 1960’s with Jones’ work on the normal Moore space problem. There are various limitations on the kind of spaces that can be obtained this way, for instance, Nyikos proved that for every tree $T$, if $X_T$ is normal, then it is also countably metacompact (CMC), i.e., there are no Dowker trees. Throughout the years, many consistency results were proven concerning the topological characteristics of spaces of the form $X_T$, but we couldn’t find similar results dealing with the spaces’ square. Here, we present a consistent construction of an Aronszajn tree $T$ such that $X_T$ is perfectly normal but $(X_T)^2$ it not even CMC. A key component of the construction is the use of `elevators’ – a device that enables to construct the tree level-by-level while optimally controlling features of its powers.

The paper is available on my webpage, as well as in a published form.