Definition 1.2.5 (Cofinal). Let $(I, \lesssim)$ be a upward/downward directed set, then $J \subset I$ is cofinal if for every $\alpha \in I$, there exists $\beta \in J$ with $\beta \gtrsim \alpha$/$\beta \lesssim \alpha$.
Definition 1.2.5 (Cofinal). Let $(I, \lesssim)$ be a upward/downward directed set, then $J \subset I$ is cofinal if for every $\alpha \in I$, there exists $\beta \in J$ with $\beta \gtrsim \alpha$/$\beta \lesssim \alpha$.