#forcing #definition #forcing_property # Definition A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ is *$\kappa$-closed* if for every $\lambda\leq\kappa$, every descending sequence $p_{0} ≥ p1 ≥ . . . ≥ p_{\alpha} ≥ . . .$ ($\alpha<\lambda$) has a lower bound. $P$ is lt;\kappa$-closed if it is $\lambda$-closed for all $\lambda<\kappa$. ^8859e0 - $\omega$-closed is also called *countably closed* or *$\sigma$-closed*^c-closed # Properties A $\kappa$-closed forcing: - preserves cardinals $\leq \kappa ^{+}$ - doesn't add new sets of size $\kappa$ A lt;\kappa$-closed forcing: - preserves cardinals $\leq \kappa$ - doesn't add new sets of size lt;\kappa$ - $\sigma$-closed forcing preserves [[Club sets and stationary sets#Club and stationary subsets of $P_{ omega_{1}}(X)$|stationary subsets]] of $\mathcal{P}_{\omega _{1}}(X)$. - ![[Velleman - Morasses, diamond, and forcing#^d817b6]] - ![[Komjáth - The Colouring Number#^030680]] # Relations ## Implied by - [[Directed closed]] ## Implies - [[Strategically closed]] - [[Distributivity]] - [[Cardinal preservation]] up to $\kappa$. - Countably closed posets are [[Proper forcing|proper]]