Vector matroid with infinite ground set?

Question

One type of matroid is called vector matroid which has finite ground set.

I am wondering is there a name for the following type of matroid, whose ground set is $\mathbb{R}^n$, and whose independent sets are sets of independent vectors?

Answer 1

There is not a single concept of matroids with infinite ground sets.

A finitary matroid is a pair $(X,\mathcal I)$ where $X$ is an infinite ground set and $\mathcal I$ is a collection of subsets of $X$ such that, in addition to the (finite) matroid axioms, it also satisfies the following axiom: if $J\subseteq X$ is infinite and every finite subset of $J$ is in $\mathcal I$, then $J\in\mathcal I$.

This additional axiom is a very literal generalization of what it means for an infinite set of vectors to be linearly independent, namely that every finite subset is linearly independent.

To answer your question, a finitary matroid whose ground set is $\mathbb R^n$ and whose independet sets are sets of linearly independent vectors is called a linear finitary matroid.

Oxley calls finitary matroids independence spaces, see the paragraph before example 3.1.2 in his book: https://www.math.lsu.edu/~oxley/infinitematroidschapter.pdf

However, the issue with finitary matroids is that duality has to be abandoned.

In Chapter 3.2 of the same link above you can find a discussion about a different class of infinite matroids called $B$-matroids that generalise finitary matroids. Oxley proved that any axiomatic system for infinite matroids that respects duality has to capture $B$-matroids as well.

For some modern developments, the paper Axioms for infinite matroids by Bruhn, Diestel, Kriesell, Pendavingh and Wollan gives an axiomatic framework for infinite matroids with duality.