The dispersion of electron cyclotron waves in a relativistic Maxwellian plasma is investigated. It is shown that the apparently very complicated picture of the coupling of the extraordinary (X) mode to Bernstein waves can be accounted for in a simple way by considering the refractive indices as Riemann-like surfaces in the Clemmow-Mullaly-Allis (CMA) parameter space, (wp2/w2, we/w), and by introducing a few topological concepts from the analysis of complex functions. A detailed study is made of the surface representing the X mode for perpendicular propagation, with special attention given to the connection between this mode and Gross-Bernstein modes. For perpendicular propagation non-transcendental approximations to the relativistic refractive indices for X and O modes can be given. We show that these approximations are good up to ~25keV and that at frequencies up to the second harmonic of the electron cyclotron frequency the X mode approximation also accounts correctly for the connection of the X mode to Bernstein modes. The accuracy and the numerical efficiency of the approximations make them well suited for routine calculations for millimetre wave applications in fusion plasmas, including the analysis of X mode and O mode reflectometry.