The ∗-Prikry condition

In this paper we isolate a property for forcing notions, the *-Prikry condition, that is similar to the Prikry condition but that is topological: A forcing P satisfies it iff for every p ∈ P and for every open dense D ⊆ P, there are n ∈ ω and q ≤∗ p such that for any r ≤ q with l(r) = l(q) + n, r ∈ D, for some length notion l. This is implicit in many proofs in literature. We prove this for the tree Prikry forcing and the long extender Prikry forcing.