In mathematics, the **Lyusternik-Schnirelmann category** of a topological space *X* is the topological invariant defined as the smallest cardinality of an open covering of *X* by contractible subsets. For example, if *X* is the circle, this takes the value two.

In general it is not so easy to compute this invariant, which was initially introduced by Lazar' Aronovich Lyusternik (1900-1981) and Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. It was, as originally defined for the case of *X* a manifold, the lower bound for the number of critical points a Morse function on *X* could possess (cf. Morse theory).