Lusternik–Schnirelmann category

Remarks

This is the normalized convention; some authors count the open sets instead, which adds $1$. If $X$ is a path-connected space, $R$ is a commutative ring, and $H^*(X; R)$ has $m$ elements of positive degree with nonzero cup product, then $\operatorname{cat}(X) \ge m$. The category is a homotopy invariant.

Example: for $n \ge 1$, $\operatorname{cat}(S^n) = 1$. The sets $U_0 = S^n \setminus \set{N}$ and $U_1 = S^n \setminus \set{S}$, where $N$ and $S$ are the poles, cover $S^n$ and are homeomorphic to $\R^n$, hence contractible, hence contractible in $S^n$. And $\operatorname{cat}(S^n) \ne 0$, since the identity of $S^n$ is not null-homotopic.

Non-example: the cover of $S^1$ by the single open set $U_0 = S^1$ does not show $\operatorname{cat}(S^1) = 0$, since the inclusion $S^1 \to S^1$ is the identity, which is not null-homotopic.

Other values: the torus $T^n = (S^1)^n$ and the real projective space $\R P^n$ both have category $n$.