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  "body": "\n## Summary\n\nA topological space has the fixed-point property if every continuous map from the space to itself fixes at least one point.\nIt is a topological invariant, so it can tell spaces apart.\nThe Brouwer fixed-point theorem says that every closed ball in $\\R^n$ has this property.\nFixed-point results of this kind are a main tool for proving that equations have solutions.\n\n## Explanations\n\n### L1 human\n\nImagine moving every point of a space to some point of the same space, without tearing it.\nThe space has the fixed-point property if, however you do this, some point stays where it was.\nThe interval $[0, 1]$ has it: the graph of a continuous map $f \\colon [0, 1] \\to [0, 1]$ must cross the diagonal line $y = x$.\nA circle does not have it, since a rotation by a small angle moves every point.\nThe real line does not have it either, since $x \\mapsto x + 1$ moves every point.\n",
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