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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $T \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nA subset $M \\subseteq X$ is a *minimal set* for $T$ if:\n\n1. $M$ is nonempty and closed.\n2. $M$ is [[invariant|WWZTW11664]]: $T(M) \\subseteq M$.\n3. No nonempty closed set $N \\subsetneq M$ satisfies $T(N) \\subseteq N$.\n\nThe system $(X, T)$ is *minimal* if $X$ itself is a minimal set.\n\n## Remarks\n\nEquivalent form: a nonempty closed invariant set $M$ is minimal if and only if, for every $x \\in M$, the forward orbit $\\set{T^n(x) : n \\ge 0}$ is [[dense|0BSJSX7ME2]] in $M$.\nIndeed, the closure of a forward orbit is closed and invariant by continuity of $T$.\nIf $X$ is nonempty and [[compact|0TVJZYB6SH]], Zorn's lemma applied to the nonempty closed invariant sets gives at least one minimal set.\nFor a continuous action of a group $G$, a minimal set is a nonempty closed $G$-invariant set with no proper nonempty closed $G$-invariant subset.\nThe action is minimal when every orbit is dense.\nExample: for $T(x) = x + \\alpha \\bmod 1$ on the circle $\\R / \\Z$ with $\\alpha$ irrational, the points $n \\alpha \\bmod 1$ are dense, so every orbit is dense and the circle is minimal.\nNon-example: for $\\alpha = 1/2$, the circle is not minimal, since the closed invariant set $\\set{0, 1/2}$ is a proper nonempty subset; clause 3 fails.\n\n## Explanations\n\n### L1 human\n\nA minimal set is closed, is mapped into itself, and cannot be shrunk to a smaller closed set with the same property.\n",
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