#definition #forcing #forcing_property # Definitions - A forcing notion $\mathbb{P}$ is $κ$-distributive if the intersection of $κ$ open dense sets is open dense. $\mathbb{P}$ is lt;κ$-distributive if it is $λ$-distributive for all $λ < κ$. - Equivalently: A forcing notion $\mathbb{P}$ is $κ$-distributive if every function $f:\kappa\to V$ in $V^{\mathbb{P}}$ is already in $V$ # Properties - A lt;\kappa$-distributive forcing preserves cardinals $\leq \kappa$ i # Relations ## Implied by - [[Directed closed]] - [[Closed forcing]] - [[Strategically closed]] ## Implies - [[Cardinal preservation]] up to $\kappa$.