  We present measurements of a topological property, the Chern number
($C_\mathrm{1}$), of a closed manifold in the space of two-level system
Hamiltonians, where the two-level system is formed from a superconducting
qubit. We manipulate the parameters of the Hamiltonian of the superconducting
qubit along paths in the manifold and extract $C_\mathrm{1}$ from the
nonadiabitic response of the qubit. By adjusting the manifold such that a
degeneracy in the Hamiltonian passes from inside to outside the manifold, we
observe a topological transition $C_\mathrm{1} = 1 \rightarrow 0$. Our
measurement of $C_\mathrm{1}$ is quantized to within 2 percent on either side
of the transition.
