 Lusternik–Schnirelmann category

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 $\mathrm{cat}(X)\ge r+1$.

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.