Net

Remarks

Example: a sequence $(x_n)_{n \ge 1}$ is a net indexed by $\N$ with its usual order, and the net converges to $x$ exactly when the sequence converges to $x$.

Example: let $A \subseteq X$ and let $x$ be in the closure of $A$. Let $D$ be the set of neighbourhoods of $x$, ordered by $U \le V$ when $V \subseteq U$; this is directed because $U \cap V$ is a neighbourhood. For each $U$ choose $x_U \in U \cap A$. Then $(x_U)$ converges to $x$: for a neighbourhood $U$ take $d_0 = U$, and every $V \ge U$ gives $x_V \in V \subseteq U$.

Non-example: in $\R$, the sequence $x_n = n$ is a net that converges to no point $x$, since the neighbourhood $(x - 1, x + 1)$ contains $x_n$ for at most two values of $n$.

A point lies in the closure of $A$ if and only if some net in $A$ converges to it. A space is Hausdorff if and only if every net in it has at most one limit.