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  "body": "\n## Summary\n\nAn absolute neighborhood retract, or ANR, is a [[metrizable space|091GJ3FYD7]] with the following property.\nWhenever it sits as a closed subset in a metrizable space, it is a retract of some neighbourhood of itself.\nANRs include manifolds, polyhedra, open subsets of Banach spaces and convex subsets of normed spaces.\nThey are the spaces with good local behaviour for homotopy theory.\nThey are locally contractible, and compact ANRs have the homotopy type of finite CW complexes.\n\n## Explanations\n\n### L1 human\n\nA retraction pushes a big space onto a subspace while keeping each point of the subspace fixed.\nPlace an ANR as a closed set inside a larger metric space.\nThen some thin open layer around it can be pushed onto it by a retraction.\nThe interval $[0, 1]$ is an ANR, and so is the circle.\nA set with infinitely many points piling up, like $\\set{0} \\cup \\set{1/n : n \\ge 1}$, is not an ANR.\nA thin layer around it in $\\R$ contains a whole interval around $0$, and this interval cannot be pushed continuously onto separate points.\n",
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