Definition 1.2.4 (Directed Set). Let $I$ be a set and $\lesssim$ be a relation on $I$, then $(I, \lesssim)$ is a directed set if

1. For any $i \in I$, $i \lesssim i$.

2. For any $i, j, k \in I$ such that $i \lesssim j$ and $j \lesssim k$, $i \lesssim k$.

and one of the following holds:

1. For any $i, j \in I$, there exists $k \in I$ with $i, j \lesssim k$.

2. For any $i, j \in I$, there exists $k \in I$ with $k \lesssim i, j$.

The directed set is upward-directed if it satisfies (3U), and downward-directed if it satisfies (3D).