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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nAn *$\\omega$-chain* in $P$ is a sequence $(x_n)_{n \\in \\N}$ with $x_0 \\le x_1 \\le x_2 \\le \\cdots$.\nThe poset $P$ is an *omega-complete partial order*, or *$\\omega$-cpo*, if every $\\omega$-chain in $P$ has a [[supremum|0R3H5PANRQ]] $\\bigvee_n x_n$ in $P$.\n\n## Remarks\n\nSome authors also require a least element.\nEvery [[directed-complete partial order|7TSXP95QN6]] is an $\\omega$-cpo, since the set of terms of an $\\omega$-chain is directed.\nExample: $\\N \\cup \\set{\\infty}$ with $n \\le \\infty$ for all $n$; an $\\omega$-chain is either eventually constant, with supremum its final value, or unbounded in $\\N$, with supremum $\\infty$.\nNon-example: $\\N$ itself, since the chain $0 \\le 1 \\le 2 \\le \\cdots$ has no supremum.\nThe converse fails for the set $\\omega_1$ of countable ordinals with their usual order.\nIt is an $\\omega$-cpo, because a countable set of countable ordinals has a countable supremum.\nBut the directed set $\\omega_1$ itself has no supremum in $\\omega_1$.\n",
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