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  "body": "\n## Summary\n\nAn upper set of a [[partially ordered set|011C29Q1S1]] is a subset that contains everything above each of its elements.\nThe dual notion is a lower set, which contains everything below each of its elements.\nUpper and lower sets are used to define order filters, the Alexandrov topology, and the lattice of down-sets of a poset.\n\n## Explanations\n\n### L1 human\n\nAn upper set is closed upwards: once an element is in, every larger element is in too.\nIn the real numbers, the interval $[3, \\infty)$ is an upper set, and so is $(3, \\infty)$.\nIn the subsets of $\\set{1, 2, 3}$ ordered by inclusion, the family of all subsets containing $1$ is an upper set.\nThe set $\\set{3}$ alone is not an upper set of $\\R$, because $4 \u003e 3$ is missing.\n",
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