Power spectra in the non-rapid eye movement sleep (NREMS) electroencephalogram (EEG) have been shown to exhibit frequency-specific topographic features that may point to functional differences in brain regions. Here, we extend the analysis to rapid eye movement sleep (REMS) and waking (W) to determine the extent to which EEG topography is determined by state under two different levels of sleep pressure. Multichannel EEG recordings were obtained from young men during a baseline night, a 40-h waking period, and a recovery night. Sleep deprivation enhanced EEG power in the low-frequency range (1–8 Hz) in all three vigilance states. In NREMS, the effect was largest in the delta band, inW, in the theta band, while in REMS, there was a peak in both the delta and the theta band. The response of REMS to prolonged waking and its pattern of EEG topography was intermediate between NREMS andW. Cluster analysis revealed a major topographic segregation into three frequency bands (1–8 Hz, 9–15 Hz, 16–24 Hz), which was largely independent of state and sleep pressure. To assess individual topographic traits within each state, the differences between pairs of power maps were compared within (i.e., for baseline and recovery) and between individuals (i.e., separately for baseline and recovery). A high degree of intraindividual correspondence of the power maps was observed. The frequency-specific clustering of power maps suggests that distinct generators underlie EEG frequency bands. Although EEG power is modulated by state and sleep pressure, basic topographic features appear to be state-independent.