Let $M$ be a matroid on ground set $E$, let $\mathcal{I}$ be its collection of independent sets, and let $\operatorname{cl}_M$ denote its closure operator. For a subset $B \subset E$, the set $B$ is a basis of $M$ if and only if $B \in \mathcal{I}$ and $\operatorname{cl}_M(B) = E$.