Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-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}}_{1leq ileq k}} of X {displaystyle X} with the property that each inclusion map U i ↪ X {displaystyle U_{i}hookrightarrow X} is nullhomotopic. For example, if X {displaystyle X} is a sphere, this takes the value two. In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-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}}_{1leq ileq k}} of X {displaystyle X} with the property that each inclusion map U i ↪ X {displaystyle U_{i}hookrightarrow X} is nullhomotopic. For example, if X {displaystyle X} is a sphere, this takes the value two.