Tree Version of Hechler Forcing

Question

In their recent (2009) paper Eventually Different Functions and Inaccessible Cardinals, Brendle and Löwe consider a 'tree version' of the Hechler forcing. This forcing $\mathbb{D}$ consists of nonempty trees $T\subseteq\omega^{<\omega}$ with the property that there is a unique stem $s\in T$ so that for every $t\in T$ extending $s$, $t\frown n\in T$ for all but finitely many $n\in\omega$. The forcing is ordered by inclusion. This forcing allows for a very elegant rank analysis, and many properties about Hechler forcing that are proved to hold using rank arguments can also be proved to hold for $\mathbb{D}$ in a conceptually simpler way. I do not know if the two notions of forcing are equivalent, and based on what is said in the paper I suspect neither do the authors. This is not my real question, though I would be happy to see an answer.

My question is simply whether this specific tree forcing has appeared anywhere in the literature previously. No reference is given in the paper but I ask because I do think I remember seeing it somewhere and now I can't seem to find mention of it in likely places.

Answer 1

This forcing is a special case of forcing with trees that branch into a filter, the filter in this case being the co-finite sets. (This, in turn, can be viewed as a special case of Shelah's creature forcing.) So the example has certainly implicitly appeared in the literature. An early reference to forcing with trees branching into filters is "Combinatorics on ideals and forcing with trees" by Marcia Groszek in J. Symbolic Logic 52 (1987), no. 3, 582–593.