  Expanding a result of Serre on finite CW-complexes, we show that the Brauer
group coincides with the cohomological Brauer group for arbitrary compact
spaces. Using results from the homotopy theory of classifying spaces for Lie
groups, we give another proof of the result of Antieau and Williams that
equality does not hold for Eilenberg--MacLane spaces of type K(Z/nZ,2).
Employing a result of Dwyer and Zabrodsky, we show the same for the classifying
spaces BG where G is an infinite-dimensional F_p-vector space. In this context,
we also give a formula expressing phantom cohomology in terms of homology.
