This talk describes a class of matroids on the ground set of rational numbers with some interesting properties. For example:Each base has finitely many fundamental sets forming a chain under inclusion.

Every set $S$ has a closure of the form $S \cup D \cup F$, where $D$ is a Dedekind cut and $F$ is finite.

In the talk, I will show how to construct such a matroid assuming $2^{\aleph_0} = \aleph_1$, and discuss why this class of matroids could help address the infinite version of Edmonds' matroid intersection theorem.