Definition 1.2.5 (Cofinal).label 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).label 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 $\beta \gtrsim \alpha$/$\beta \lesssim \alpha$.
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