4.3 Nets

Definition 4.3.1 (Net). Let $X$ be a set and $A$ be an upward-directed set, then a net in $X$ is a mapping $A \to X$, denoted $\net{x}\subset X$.

Definition 4.3.2 (Eventually). Let $X$ be a set, $\net{x}\subset X$, and $E \subset X$, then $\net{x}$ is eventually in $E$ if there exists $\alpha_{0} \in A$ such that $x_{\alpha} \in E$ for all $\alpha \gtrsim \alpha_{0}$.

Definition 4.3.3 (Frequently). Let $X$ be a set, $\net{x}\subset X$, and $E \subset X$, then $\net{x}$ is frequently in $E$ if for all $\alpha_{0} \in A$, there exists $\alpha \gtrsim \alpha_{0}$ such that $x_{\alpha} \in E$.

Definition 4.3.4 (Convergence). Let $X$ be a topological space and $\net{x}\subset X$, then $\net{x}$ converges to $x$ if for every $U \in \cn(x)$, $\net{x}$ is eventually in $U$, denoted $x_{\alpha} \to x$.