Are topological matroids something that has been studied?
I would like to construct and study mathematical models of an abstract world consisting of a finite number of particles in a finite number of states of motion and think that the "nearby" from topology and the independence from matroids together could be a good start.
The idea is to derive a space-time like background from structures on a finite set of states of motion. I want to examine possibilities to study sets of moving elements (i.e. elements that would be interpreted as moving in a geometrical frame) represented as "states" in a world without a continuum. As an abstraction, a sort of game with structures...
With a topological matroid I mean a triplet $(X,\mathcal I,\tau)$, where $(X,\tau)$ is a finite topological space, $(X,\mathcal I)$ is a matroid and also eventually some interaction between the structures.
All tips and ideas is welcome.