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If $E$ is a finite set and $k \in \mathbb{N}$, then the uniform matroid of rank $k$ is defined as the matroid generated by taking the collection of $k$ element subsets of $E$ as a basis. Is there an analog of this notion when $E$ is not finite?

Currently reading this paper, other references or examples would be great.

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Browsing the Wikipedia article suggests that there does not seem to be an agreed-upon definition of an infinite matroid. What definition are you using? – Qiaochu Yuan Jun 15 '12 at 23:10
@Qiaochu: The purpose of the cited paper is to give an axiomatic definition of infinitary matroids with duality; it actually gives five equivalent definitions, in terms of independent sets, bases, closure, circuits, and rank. It turns out that cardinality is too crude a measure, so a notion of relative rank is used that reduces to the usual notion in the finite case. – Brian M. Scott Jun 15 '12 at 23:37

An uniform matroid is a matroid such that: $\forall I\in \mathcal{I}$, $\forall x\in I$, $\forall y\in E\backslash I$, $I-x+y\in \mathcal{I}$. Here some slides of a talk about the implications of the existance of infinite uniform matroids,

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