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  "body": "\n## Summary\n\nA [[filter|15PF73BKY6]] on a [[topological space|0CQHFPGJ45]] converges to a point $x$ when it contains every [[neighbourhood|0ZCY0Z80B6]] of $x$.\nThis notion extends the [[convergence of sequences|0W4CN9E2W5]] to all spaces.\nClosures, [[continuity|0VGFHTM75Y]], [[compactness|0TVJZYB6SH]] and the [[Hausdorff property|0S1FJ8VEX9]] can all be described by convergent filters.\n\n## Explanations\n\n### L1 human\n\nA filter is a family of \"big\" sets, and it converges to $x$ when every set around $x$, however small, counts as big.\nFor a sequence $x_1, x_2, \\dots$, take as big sets those that contain all terms from some index on.\nThis filter converges to $x$ exactly when the sequence converges to $x$.\nIn spaces where sequences do not see the whole topology, filters still do.\n",
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