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).