Closure, distributivity, and chain conditions for forcing orders

Let $\kappa$ be an infinite regular cardinal and let $P$ be a nonempty forcing preorder, with stronger conditions smaller. The order is $\kappa$-closed if every descending sequence $⟨p_{\xi }:\xi <\gamma ⟩$ of length $\gamma <\kappa$ has a common lower bound. It is $\kappa$-distributive if the intersection of every family of fewer than $\kappa$ dense open subsets of $P$ is dense. It is $\kappa$-cc if every antichain has cardinality below $\kappa$.

Thus ccc is ${\mathrm{\aleph }}_1$-cc. “Countably closed” or “$\sigma$-closed” means ${\mathrm{\aleph }}_1$-closed: all countable descending chains have lower bounds. It does not mean $\omega$-closed, which only addresses finite chains and is automatic for a preorder with its last member as a lower bound.