Notes on the topology of independence structures

Following Welsh, a pre-independence space (pi-space) is a set $M$ together with a non-empty collection $I(M)$ of subsets of $M$, called independent sets, which is closed under taking subsets, and finite independent sets satisfy the exchange property from matroid theory. We show that $I(M)$, viewed as a poset, is contractible if it is infinite-dimensional, and Cohen-Macaulay otherwise. Moreover, the proper part of the associated poset of flats is also contractible in the infinite-dimensional case, and Cohen-Macaulay otherwise. These results generalize those for independence complexes and geometric lattices of (finite) matroids.