Net Convergence

Last updated Nov 1, 2022

Let $X$ be a Topological Space and ${x_{\alpha}}{\alpha \in A} \subset X$ be a Net. We say $x{\bullet}$ converges to $x \in X$ if for all Open $U \subset X$ so $U \ni x$, there exists $\alpha_{0} \in A$ so that for all $\alpha \geq \alpha_{0}$, $x_{\alpha} \in U$. If this holds, we write $$\lim\limits_{\alpha \in A} x_{\alpha} = x \text{ or } x_{\alpha} \to x.$$