Many studies of brain function with positron emission tomography (PET) involve the interpretation of a subtracted PET image, usually the difference between two images under baseline and stimulation conditions. The purpose of these studies is to see which areas of the brain are activated by the stimulation condition. In many cognitive studies, the activation is so slight that the experiment must be repeated on several subjects and the subtracted images are averaged to improve the signal to noise ratio. The averaged image is then standardized to have unit variance and then searched for local maxima (Fox et al., 1988). The main problem facing investigators is which of these local maxima are statistically significant. We describe a simple method for determining an approximate $p$-value for the global maximum based on the theory of Gaussian random fields as developed by Adler and Hasofer (1976) and Adler (1981). The $p$-value is proportional to the volume searched divided by the product of the FWHMs of the image reconstruction process, or number of resolution elements (resels). Rather than working with local maxima as in Fox et al. (1988), our method focuses on the Euler characteristic of the set of voxels with a value larger than a given threshold. The Euler characteristic depends only on the topology of the regions of high activation, irrespective of their shape. For large threshold values this is approximately the same as the number of isolated regions of activation above the threshold. We can thus determine not only if any activation has taken place, but we can also estimate how many isolated regions of activation are present.