TITLE: Net convergence in metric spaces
QUESTION [5 upvotes]: This is a question about convergence of nets which I don't quite understand yet. In metric spaces convergence of sequences encodes the topology but suppose we want to study convergence of nets even though. When can we pass to countable subnets? In other words,

Given a net $(x_\lambda)_{\lambda\in \Lambda}$ in a separable metric space $X$ that converges to some $x\in X$. Can we find a countable subnet $\Lambda^\prime \subset \Lambda$ such that $(x_\lambda)_{\lambda\in \Lambda^\prime}$ converges to $x$?

REPLY [6 votes]: You don’t even need separability.
Let $\langle X,d\rangle$ be a metric space, and suppose that $\langle x_\lambda:\lambda\in\Lambda\rangle$ is a net in $X$ converging to $x\in X$. For each $n\in\Bbb N$ there is a $\lambda_n\in\Lambda$ such that $d(x,x_\lambda)<2^{-n}$ whenever $\lambda\in\Lambda$ and $\lambda_n\preceq\lambda$. Since $\langle\Lambda,\preceq\rangle$ is directed, we may further assume that $\lambda_m\preceq\lambda_n$ whenever $m\le n$. Now let $\Lambda'=\{\lambda_n:n\in\Bbb N\rangle$; then $\langle x_\lambda:\lambda\in\Lambda'\rangle=\langle x_{\lambda_n}:n\in\Bbb N\rangle$ converges to $x$.
(Note, however, that $\langle x_\lambda:\lambda\in\Lambda'\rangle$ is not necessarily a subnet of $\langle x_\lambda:\lambda\in\Lambda\rangle$ by either of the definitions of subnet given in the Wikipedia article or by the better definition used in J.F. Aarnes & P.R. Andenaes, ‘On Nets and Filters’, Mathematica Scandinavica $31$ ($1972$), $285$-$292$.)