> [!quote] Idea > > A simpler subcollection of a directed set, which allows inferring properties of elements in the directed sets using elements that come after them. > [!definition] > > Let $(A, \lesssim)$ be a [[Directed Set|directed set]]. A subset $B \subset A$ is **cofinal** if for any $\alpha \in A$, there exists $\beta \in B$ such that $\beta \gtrsim \alpha$. > [!theorem] > > Let $\net{x}$ be a [[Net|net]], then $x_\alpha \to x$ if and only for any cofinal subset $B \subset A$, there exists a cofinal subset $C \subset B$ such that $\angles{x_\gamma}_{\gamma \in C}$ converges to $x$. > > *Proof*. If $x_\alpha \to x$, then $\netb{x}$ converges to $x$ for all $B \subset A$ cofinal, and $\angles{x_\gamma}_{\gamma \in C}$ converges to $x$ for all $C \subset B$ cofinal. > > Suppose that the subnet condition holds, but $x_\alpha \not\to x$, then there exists $U \in \cn^o(x)$ such that $x_\alpha \not\in U$ frequently. This induces a cofinal subset $B = \bracs{\alpha: x_\alpha \not\in U}$, which cannot converge to $x$.