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  "body": "\n## Definition\n\nThe axiom of dependent choice is the following statement.\nLet $X$ be a nonempty [[set|0SJ3RX9TXA]] and let $R$ be a [[binary relation|08EJC6Q25H]] on $X$.\nAssume that for each $x \\in X$ there is $y \\in X$ with $x \\mathrel{R} y$.\nThen there is a [[sequence|0QE8STGJH2]] $(x_n)_{n \\in \\N}$ of elements of $X$ such that $x_n \\mathrel{R} x_{n+1}$ for each $n \\in \\N$.\n\n## Remarks\n\nExample: let $X = \\N$ and let $m \\mathrel{R} n$ mean $m \u003c n$.\nEach $m$ has $m + 1$ with $m \\mathrel{R} (m+1)$, and the sequence $x_n = n$ satisfies the conclusion.\nNon-example of the hypothesis: on $X = \\set{0, 1}$ with $R = \\set{(0, 1)}$, the element $1$ has no $y$ with $1 \\mathrel{R} y$.\nNo sequence with $x_n \\mathrel{R} x_{n+1}$ for all $n$ exists, so the hypothesis cannot be dropped.\nThe axiom is abbreviated DC.\nOver ZF, the [[axiom of choice|0HTGB13FE2]] implies DC, and DC implies the axiom of countable choice; neither implication reverses.\nOver ZF, DC is equivalent to the [[Baire category theorem|0A7KGAYHGV]] for complete metric spaces.\nAn equivalent form says that a relation $R$ on a nonempty set is [[well-founded|0230ECX1Q5]] if and only if there is no sequence with $x_{n+1} \\mathrel{R} x_n$ for all $n$.\n",
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