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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nFor $x, y \\in P$, the closed interval from $x$ to $y$ is $[x, y] = \\set{z \\in P : x \\le z \\le y}$.\nThe poset $P$ is *locally finite* when $[x, y]$ is a [[finite set|070R982G4B]] for all $x, y \\in P$.\n\n## Remarks\n\nExample: $\\N^k$ with the product order is locally finite.\nCheck: if $x \\le y$, the interval $[x, y]$ is the box $\\prod_{i=1}^k \\set{x_i, \\ldots, y_i}$, which has $\\prod_i (y_i - x_i + 1)$ elements; if $x \\not\\le y$, it is empty.\n\nNon-example: $\\Q$ with the usual order is not locally finite, since $[0, 1]$ is infinite.\n\nEvery finite poset is locally finite.\nThe interval $[x, y]$ is empty unless $x \\le y$, so only comparable pairs matter.\nIn Mathlib the class `LocallyFiniteOrder` provides the intervals as finite sets.\n",
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