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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]].\nThe map $f$ is *topologically transitive* if for every pair of nonempty [[open sets|02BH3MCBQB]] $U, V \\subseteq X$ there is an integer $n \\ge 1$ with $f^n(U) \\cap V \\neq \\emptyset$.\n\n## Remarks\n\nAn equivalent form is $U \\cap f^{-n}(V) \\neq \\emptyset$, since both say that some point of $U$ is sent into $V$ by $f^n$.\nLet $X$ be a compact metric space without isolated points.\nThen $f$ is topologically transitive if and only if some forward orbit $\\set{f^n(x) : n \\ge 0}$ is [[dense|0BSJSX7ME2]] in $X$.\nIn that case the points with dense forward orbit form a dense $G_\\delta$ set.\nExample: let $f(x) = x + \\alpha \\bmod 1$ on the circle $\\R / \\Z$ with $\\alpha$ irrational.\nEvery forward orbit is dense, so for nonempty open $U, V$ and a point $x \\in U$, some $f^n(x)$ with $n \\ge 1$ lies in $V$.\nNon-example: the identity map of $[0, 1]$, with $U = (0, 1/3)$ and $V = (2/3, 1)$, has $f^n(U) \\cap V = U \\cap V = \\emptyset$ for all $n$.\n\n## Explanations\n\n### L1 human\n\nA transitive map moves some point of each open region into each other open region, at some later time.\n",
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