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  "body": "\n## Summary\n\nLusternik–Schnirelmann theory gives lower bounds for the number of [[critical points|0YTCMFB4HC]] of a function from the topology of its domain, without any nondegeneracy assumption.\nIts main result is that a $C^1$ function on a closed manifold has at least $\\operatorname{cat}(M) + 1$ critical points, where $\\operatorname{cat}(M)$ is the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].\nThe critical values are obtained by minimax over closed sets of increasing category.\nIt is a branch of [[critical point theory|9ZQC7ARWTT]], complementary to Morse theory, and it extends to infinite dimensions under the [[Palais–Smale condition|4724KRNHAT]].\n\n## Explanations\n\n### L1 human\n\nMorse theory counts critical points of a function with only clean critical points, and gives bounds from the homology of the space.\nLusternik and Schnirelmann asked what can be said for any smooth function, however degenerate.\nTheir answer is a number measuring how many contractible pieces are needed to cover the space.\nFor example, a torus needs three such pieces, so every smooth function on a torus has at least three critical points.\nOn a sphere the bound is two: a maximum and a minimum.\n",
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