We present Faint Object Camera (FOC) ultraviolet images of the central 14 x 14″ of Messier 31 and Messier 32. The hot stellar population detected in the composite UV spectra of these galaxies is partially resolved into stars, and we measure their colors and apparent magnitudes. We detect 433 stars in M31 and 138 stars in M32, down to limits of mF275W = 25.5 mag and mF175W = 24.5 mag. We investigate the luminosity functions of the sources, their spatial distribution, their color-magnitude diagrams, and their total integrated far-UV flux. Although M32 has a weaker UV upturn than M31, the luminosity functions and color-magnitude diagrams of M31 and M32 are surprisingly similar, and are inconsistent with a majority contribution from any of the following: post-AGB stars more massive than 0.56 M⊙, main sequence stars, or blue stragglers. The luminosity functions and color-magnitude diagrams are consistent with a dominant population of stars evolving from the extreme horizontal branch (EHB) along tracks of mass 0.47–0.53 M⊙. These stars are well below the detection limits of our images while on the zero-age EHB, but become detectable while in the more luminous (but shorter) post-HB phases. Our observations require that only a very small fraction of the main sequence population (2% in M31 and 0.5% in M32) in these two galaxies evolve though the EHB and post-EHB phases, with the remainder rapidly evolving through bright post-AGB evolution with few resolved stars expected in the small field of view covered by the FOC.