Basis Exchange Characterization of Valuated Matroids is a result in [matroid theory](/page/Matroid%20Theory). It formalizes the assertion that let be a finite set, let be an integer with , and let \begin{align } p:\binom{E}{r}\to \mathbb{R}\cup\{\infty\} \end{align } be a function. Define \begin{align } \mathcal{B}:=\{B\in \binom{E}{r}:p(B)<\infty\}. \end{align } Then is a.