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  "body": "\n## Definition\n\nLet $M$ be a compact connected [[smooth manifold|F45VQ47GPV]] without boundary and let $f \\colon M \\to \\R$ be a $C^1$ function.\nFor a closed subset $A \\subseteq M$, let $\\operatorname{cat}_M(A)$ be the least $m \\ge 0$ such that $m + 1$ open subsets of $M$, each contractible in $M$, cover $A$.\nThen $\\operatorname{cat}_M(M) = \\operatorname{cat}(M)$, the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].\nFor $k = 0, 1, \\dots, \\operatorname{cat}(M)$, the *Lusternik–Schnirelmann minimax values* of $f$ are\n$$c_k = \\inf \\set{ \\max_{x \\in A} f(x) : A \\subseteq M \\text{ closed}, \\ A \\ne \\emptyset, \\ \\operatorname{cat}_M(A) \\ge k }.$$\n*Lusternik–Schnirelmann theory* is the [[critical point theory|9ZQC7ARWTT]] built on these values.\nIts basic theorem states:\n\n1. each $c_k$ is a critical value of $f$, and $c_0 \\le c_1 \\le \\cdots \\le c_{\\operatorname{cat}(M)}$;\n2. if $c_k = c_{k+1}$ for some $k$, then $f$ has infinitely many [[critical points|0YTCMFB4HC]] at that level;\n3. hence $f$ has at least $\\operatorname{cat}(M) + 1$ critical points.\n\n## Remarks\n\nThe proof uses a deformation along a pseudo-gradient flow.\nIf $c$ is not a critical value, the sublevel set $\\set{f \\le c + \\varepsilon}$ can be pushed into $\\set{f \\le c - \\varepsilon}$, and such deformations do not increase category.\nIf the critical set at level $c_k = c_{k+1}$ were finite, it would have a neighbourhood of category $0$, and subadditivity of category would give a contradiction.\nSince one-point sets are allowed for $k = 0$, the first value is $c_0 = \\min f$.\n\nExample: for $M = T^2$, $\\operatorname{cat}(T^2) = 2$, so every $C^1$ function on the torus has at least three critical points.\nThis bound is attained by suitable functions, while a Morse function on $T^2$ has at least four.\n\nNon-example: the theory does not apply as stated to $f(x) = \\arctan x$ on the non-compact manifold $\\R$.\nThere $\\operatorname{cat}(\\R) = 0$ would predict one critical point, but $f$ has none, since the infimum defining $c_0$ is not attained.\nIn infinite dimensions compactness of $M$ is replaced by the [[Palais–Smale condition|4724KRNHAT]] together with completeness and a lower bound on $f$.\n",
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