In mathematics, the Lyusternik–Schnirelmann category of a topological space X {\displaystyle X} is the homotopy invariant defined to be the smallest integer number k {\displaystyle k} such that there is an open covering { U i } 1 ≤ i ≤ k {\displaystyle \{U_{i}\}_{1\leq i\leq k}} of X {\displaystyle X} with the property that each inclusion map U i ↪ X {\displaystyle U_{i}\hookrightarrow X} is nullhomotopic. Wikipedia