Lusternik–Schnirelmann category

Summary

The Lusternik–Schnirelmann category of a space $X$ is the least number of open sets, minus one, that cover $X$ and are each contractible inside $X$. It is a homotopy invariant that measures how far a space is from being contractible. Lusternik and Schnirelmann introduced it to bound from below the number of critical points of a smooth function on a closed manifold. It is itself bounded below by the cup-length of the cohomology ring.