Cup length
= Cup length
{title2=$\operatorname{cup}_R(X)$}
The largest number of positive-degree <cohomology classes> whose <cup product> is nonzero, with a specified coefficient ring $R$; it can be infinite. It is zero if there is no nonzero positive-degree class. The <vanishing cup product from an open cover> argument implies that <Lusternik–Schnirelmann category> is at least cup length plus one. For <Complex projective space> $\mathbb{CP}^r$, its integral cup length is $r$.