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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nAn open subset $U \\subseteq X$ is *contractible in $X$* if the inclusion $U \\to X$ is [[null-homotopic|SMEV5BSQ5S]].\nThe *Lusternik–Schnirelmann category* $\\operatorname{cat}(X)$ is the least integer $m \\ge 0$ such that there is an [[open cover|049PXYHW1T]] $X = U_0 \\cup U_1 \\cup \\cdots \\cup U_m$ by $m + 1$ open sets, each contractible in $X$.\nIf no such finite cover exists, $\\operatorname{cat}(X) = \\infty$.\n\n## Remarks\n\nThis is the normalized convention; some authors count the open sets instead, which adds $1$.\nIf $X$ is a path-connected space, $R$ is a commutative ring, and $H^*(X; R)$ has $m$ elements of positive degree with nonzero cup product, then $\\operatorname{cat}(X) \\ge m$.\nThe category is a homotopy invariant.\n\nExample: for $n \\ge 1$, $\\operatorname{cat}(S^n) = 1$.\nThe sets $U_0 = S^n \\setminus \\set{N}$ and $U_1 = S^n \\setminus \\set{S}$, where $N$ and $S$ are the poles, cover $S^n$ and are homeomorphic to $\\R^n$, hence contractible, hence contractible in $S^n$.\nAnd $\\operatorname{cat}(S^n) \\ne 0$, since the identity of $S^n$ is not null-homotopic.\n\nNon-example: the cover of $S^1$ by the single open set $U_0 = S^1$ does not show $\\operatorname{cat}(S^1) = 0$, since the inclusion $S^1 \\to S^1$ is the identity, which is not null-homotopic.\n\nOther values: the torus $T^n = (S^1)^n$ and the real projective space $\\R P^n$ both have category $n$.\n",
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