  Topological invariants, such as the Chern number, characterise topological
phases of matter. Here we provide a method to detect Chern numbers in systems
with two distinct species of fermion, such as spins, orbitals or several atomic
states. We analytically show that the Chern number can be decomposed as a sum
of component specific winding numbers, which are themselves physically
observable. We apply this method to two systems, the quantum spin Hall
insulator and a staggered topological superconductor, and show that (spin)
Chern numbers are accurately reproduced. The measurements required for
constructing the component winding numbers also enable one to probe the
entanglement spectrum with respect to component partitions. Our method is
particularly suited to experiments with cold atoms in optical lattices where
time-of-flight images can give direct access to the relevant observables.
