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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\n$X$ has the *fixed-point property* if for every [[continuous map|0X3WGX95FE]] $f \\colon X \\to X$ there is a point $x \\in X$ with $f(x) = x$.\n\n## Remarks\n\nExample: the interval $[0, 1]$ has the fixed-point property.\nFor continuous $f \\colon [0, 1] \\to [0, 1]$, the function $g(x) = f(x) - x$ satisfies $g(0) \\ge 0$ and $g(1) \\le 0$, so $g$ has a zero by the [[intermediate value theorem|00KAFBR32E]].\nNon-example: the [[circle|060W5CBZP6]] $S^1 \\subseteq \\C$ fails it, because the rotation $z \\mapsto -z$ is continuous and has no fixed point.\nIf $X$ has the fixed-point property and $Y$ is homeomorphic to $X$, then $Y$ has it as well.\nA retract of a space with the fixed-point property also has it.\nThe Brouwer fixed-point theorem states that the closed unit ball of $\\R^n$ has the fixed-point property.\n",
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