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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nA set $W \\subseteq X$ is a *wandering set* for $f$ if $f^n(W) \\cap W = \\emptyset$ for every integer $n \\ge 1$.\nA point $x \\in X$ is a *wandering point* if it has an [[open|02BH3MCBQB]] neighbourhood $U$ that is a wandering set.\n\n## Remarks\n\nIf $f$ is a [[homeomorphism|0QT00PB5G2]], the images $f^n(W)$, $n \\in \\Z$, of a wandering set are pairwise disjoint.\nThe set of wandering points is open.\nIn measurable dynamics, let $T$ be a measurable map of a [[measure space|P15VQS5DJ9]] $(X, \\mathcal{A}, \\mu)$.\nA measurable set $W$ of positive measure is called wandering when the sets $T^{-n}(W)$, $n \\ge 0$, are pairwise disjoint.\nExample: for $f(x) = x + 1$ on $\\R$, the set $W = (0, 1)$ has $f^n(W) = (n, n+1)$, which is disjoint from $W$ for $n \\ge 1$; so every point of $\\R$ is wandering.\nNon-example: for $f(x) = x/2$ on $\\R$, any open interval $U$ containing $0$ satisfies $0 \\in f^n(U) \\cap U$ for all $n$, so $U$ is not wandering and $0$ is not a wandering point.\n\n## Explanations\n\n### L1 human\n\nA wandering set never overlaps with any of its later images.\nA wandering point is a point with a small open neighbourhood that wanders away and never comes back to meet itself.\n",
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