Lusternik–Schnirelmann category ($\mathrm{cat}(X)$)

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

Every smooth real-valued function on a closed connected smooth manifold has at least its unnormalized Lusternik–Schnirelmann category many distinct critical points. No nondegeneracy hypothesis is required. If there are infinitely many critical points the bound is immediate; in the finite case, the sublevel-set deformation argument shows that category can increase only at critical levels, by at most the number of critical points on that level. In conjunction with cup length, this gives at least $r+1$ critical points whenever there is a nonzero $r$-fold positive-degree cup product. This theorem is distinct from the antipodal covering theorem also sometimes named the Lusternik–Schnirelmann theorem. A primary account of the critical-point formulation is Conley pairs in geometry—Lusternik–Schnirelmann theory and more.