It needs a way to measure distances, so it's a metric space. Does that make sense?
I'm a physicist with less knowledge about math than I'd like, so forgive me if the question sounds a bit vague. I know that in physics we usually deal with smooth manifolds, a trivial euclidean space in classical physics and curved ones in General Relativity (possibly with non trivial topologies) etc. But when we make simulations in a discrete grid, we clearly have a fundamentally different structure! So what is this structure? I know it must be a topological space, it's essentially a graph, after all, but that's not saying much either :P I'm also interested in knowing what connects these seemingly different structures together.. For example, how is the manifold dimension characterized in its discrete version? (my intuition suggests it has to do with the number of connections each node has)