Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space ${\displaystyle X}$ is the homotopical invariant defined to be the smallest integer number ${\displaystyle k}$ such that there is an open covering ${\displaystyle \{U_{i}\}_{1\leq i\leq k}}$ of ${\displaystyle X}$ with the property that each inclusion map ${\displaystyle U_{i}\hookrightarrow X}$ is nullhomotopic. For example, if ${\displaystyle X}$ is a sphere, this takes the value two.

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