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 ∈Pand for every open dense D ⊆P, there are n ∈ωand 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.
