The Lusternik–Schnirelmann (LS) category is a concept in algebraic topology that measures the "complexity" of a topological space in terms of how it can be covered by open sets that have some sort of "simple" structure, specifically, contractible sets.
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Use the unnormalized convention: the least number of open subsets covering whose inclusions into are null-homotopic. The subsets need not themselves be contractible. This number is one for a nonempty contractible space. If positive-degree classes have nonzero product, a cover by categorical sets would make that product zero by relative cup products; hence .