#definition #forcing #forcing_property # Definition A notion of forcing $\mathbb{P}=(P, <)$ is $λ$-directed closed if whenever $D ⊂ P$ is such that $|D| ≤ λ$ and for any $d_1 , d_2 ∈ D$ there is some $e ∈ D$ with $e ≤ d_1$ and $e ≤ d_2$ , then there exists a $p ∈ P$ such that $p ≤ d$ for all $d ∈ D$. I.e. every directed set of size $\leq \lambda$ has a lower bound. # Properties ## Preservation under iteration **Lemma (Jech 21.7. pg. 395)** 1. If $\mathbb{P}$ is $λ$-directed closed, and if $\Vdash_{P}\dot{Q}$ is $λ$-directed closed, then $P ∗ \dot{Q}$ is $λ$-directed closed. 2. If $\mathrm{cf} α > λ$, if $\mathbb{P}_α$ is a direct limit and if for each $β < α$, $\mathbb{P}_β$ is $λ$-directed closed, then $\mathbb{P}_α$ is $λ$-directed closed. 3. Let $\mathbb{P}_α$ be a forcing iteration of $\langle\dot{Q}_β : β < α\rangle$ such that for each limit ordinal $β ≤ α$, $P_β$ is either a direct limit or an inverse limit. Assume that for each $β < α$, $\dot{Q}_β$ is a $λ$-directed closed forcing in $V^{\mathbb{P}_β}$ . If for every limit ordinal $β ≤ α$ such that $\mathrm{cf} β ≤ λ$, $\mathbb{P}_β$ is an inverse limit, then $\mathbb{P}_α$ is $λ$-directed closed. # Relations ## Implied by - [[]] ## Implies - [[Closed forcing]] - [[Strategically closed]] - [[Distributivity]] - [[Cardinal preservation]] up to $\kappa$.