The achievable spatial resolution and the contrast transfer function (CTF) are key parameters characterizing an X-ray microscope. We measured the spatial resolution and the contrast transfer function of the transmission X-ray microscope (TXM) at the electron storage ring BESSY II. The TXM uses the radiation of an undulator source and operates under partially coherent illumination conditions. For spatial resolutions down to 25 nm, our measurements of the CTF's are in good agreement with theoretical CTF data for partial coherence. With higher resolution zone plate objectives, we measured a spatial resolution (half-pitch) of 11 nm in 1st and 3rd order of diffraction. However, with these objectives the stray light level increases significantly.