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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], let $\\mathcal{F}$ be a [[filter|15PF73BKY6]] on $X$, and let $x \\in X$.\nLet $\\mathcal{N}(x)$ be the filter of [[neighbourhoods|0ZCY0Z80B6]] of $x$.\nThe filter $\\mathcal{F}$ *converges* to $x$, written $\\mathcal{F} \\to x$, if $\\mathcal{N}(x) \\subseteq \\mathcal{F}$.\nThen $x$ is a *limit* of $\\mathcal{F}$.\n\n## Remarks\n\nFor a map $f \\colon S \\to X$ from a set $S$ and a filter $\\mathcal{G}$ on $S$, the *image filter* is $f(\\mathcal{G}) = \\set{B \\subseteq X : f^{-1}(B) \\in \\mathcal{G}}$.\nOne says that $f(s) \\to x$ along $\\mathcal{G}$ if $f(\\mathcal{G}) \\to x$.\n\nExample: let $(x_n)$ be a sequence in $X$, and let $\\mathcal{F}$ be the family of sets $B \\subseteq X$ that contain $x_n$ for all large $n$.\nThen $\\mathcal{F}$ is a filter, and $\\mathcal{F} \\to x$ if and only if the sequence [[converges|0W4CN9E2W5]] to $x$, since both mean that each neighbourhood of $x$ contains $x_n$ for all large $n$.\n\nExample: the neighbourhood filter $\\mathcal{N}(x)$ converges to $x$, and so does the principal filter $\\set{B : x \\in B}$.\n\nNon-example: in $\\R$, let $\\mathcal{F}$ be the filter of sets that contain an interval $[n, \\infty)$.\nFor each $x \\in \\R$, the neighbourhood $(x - 1, x + 1)$ contains no such interval, so $\\mathcal{F}$ does not converge to $x$.\n\nA space is [[Hausdorff|0S1FJ8VEX9]] if and only if every filter on it has at most one limit.\n",
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