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  "body": "\n## Definition\n\nLet $(D, \\le)$ be a directed set and let $X$ be a set.\nA *net* in $X$ indexed by $D$ is a map $D \\to X$, written $(x_d)_{d \\in D}$.\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $x \\in X$.\nThe net $(x_d)_{d \\in D}$ *converges* to $x$ if for every [[neighbourhood|0ZCY0Z80B6]] $U$ of $x$ there is $d_0 \\in D$ such that $x_d \\in U$ for all $d \\in D$ with $d \\ge d_0$.\n\n## Remarks\n\nExample: 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|0W4CN9E2W5]] to $x$.\n\nExample: let $A \\subseteq X$ and let $x$ be in the [[closure|0QEAJB91GZ]] of $A$.\nLet $D$ 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.\nFor each $U$ choose $x_U \\in U \\cap A$.\nThen $(x_U)$ converges to $x$: for a neighbourhood $U$ take $d_0 = U$, and every $V \\ge U$ gives $x_V \\in V \\subseteq U$.\n\nNon-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 $n$.\n\nA point lies in the closure of $A$ if and only if some net in $A$ converges to it.\nA space is [[Hausdorff|0S1FJ8VEX9]] if and only if every net in it has at most one limit.\n",
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