Motivated by homotopy type theory, but using traditional topological terminology, I will illustrate how taking seriously the idea of a "space of spaces" leads to new insights into classical topics. I'll begin by discussing "central" spaces, and how they can be used to give new models of Eilenberg-Mac Lane spaces whose elements are themselves oriented Eilenberg-Mac Lane spaces. Using these, we can give a new description of the Euler class of an oriented sphere bundle, and a concrete description of the cup product operation in integral cohomology. The take-away message will be about the general techniques, rather than these specific results, with the hope that the use of universes in topology will spread.