= Lusternik–Schnirelmann category
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{title2=$\operatorname{cat}(X)$}
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= LS category
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{synonym}
Use the unnormalized convention: the least number of open subsets covering $X$ whose inclusions into $X$ are <null-homotopic>. The subsets need not themselves be contractible. This number is one for a nonempty contractible space. If $r$ positive-degree classes have nonzero product, a cover by $r$ categorical sets would make that product zero by relative <cup products>; hence $\operatorname{cat}(X)\geq r+1$.
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