# Definition A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ has the *$\kappa$-chain condition* ($\kappa$-c.c.) if every antichain has size at most $\kappa$. ^kcc - The $\omega_{1}$-c.c. is also called the *countable chain condition* (c.c.c.). ^ccc # Properties - If $\mathbb{P}$ has $\kappa$-c.c. for regular $\kappa$ then - All cofinalities (and hence all cardinals) $\geq\kappa$ are preserved. - Let $\kappa$ be a regular uncountable cardinal. Let $\mathbb{P}, \mathbb{Q}$ be forcing posets. Then $\mathbb{P} * \dot{\mathbb{Q}}$ has the $\kappa-c c$ iff $\Vdash_{\mathbb{P}}$ " $\dot{\mathbb{Q}}$ has the $\kappa-c c$ ". (See Jech 16.4 and 16.5) ^eaac13 - If $\mathbb{P}$ is a forcing notion such that $\mathbb{P}\times \mathbb{P}$ is c.c.c. then $\mathbb{P}$ cannot add an $\omega_{1}$ branch to an $\omega_{1}$ tree. # Relations ## Implied by - [[]] ## Implies - [[Cardinal preservation]] starting from $\kappa$. # Preservation ## Preserved by: - Finite support iteration ## Not necessarily preserved by: - Countable support iteration (see [[Forcing with partial functions#Curious observation]]) - Products (the product of a [[Trees|Souslin tree]] with itself is not Souslin)