Here, we formally develop theoretical methods to topologically classify a wide class of bianisotropic continuous media. It is shown that for continuous media, the underlying wave vector space may be regarded as the Riemann sphere. We derive sufficient conditions that ensure that the pseudo-Hamiltonian that describes the electrodynamics of the continuous material is well behaved so that the Chern numbers are integers. Our theory brings the powerful ideas of topological photonics to a wide range of electromagnetic waveguides and platforms with no intrinsic periodicity and sheds light over the emergence of edge states at the interfaces between topologically inequivalent continuous media.