Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless E is contained in a line (de Bruijn and Erdős, 1948). More generally, let $E$ be a spanning subset of a $d$-dimensional vector space. We show that, in the collection of subspaces spanned by subsets of $E$, there are at least as many $(d-k)$-dimensional subspaces as there are $k$-dimensional subspaces, for every $k$ at most $d/2$. This confirms the “top-heavy” conjecture of Dowling and Wilson from 1974. Joint work with Botong Wang.