I will give a topological proof that for any matroid M, the Whitney numbers of the first kind of M are greater than or equal to those of any weak map image of M, a result previously shown by Kung and Nguyen using algebraic techniques. This work utilizes a recent construction of Engstršm for obtaining topological representations of matroids via diagrams of spaces. In particular, I will show that the Whitney numbers of the first kind of M are encoded in the Betti numbers of the codimension two homotopy sphere arrangement of M and that every surjective weak map between matroids induces a surjection in homology between their topological representations.