Is the Universe finite or infinite in spatial extent, and what `shape' does it have? These fundamental questions are typically studied within the context of the standard model of cosmology where the Universe is assumed to be homogeneous and isotropic and the spacetime metric is of FLRW form. But these questions can be addressed in much more general cosmological models, with the only assumption being that the average flow of matter is irrotational. To do so in general, without a precise form of the spacetime metric, we use (smooth) Bakry-Émery generalizations of the Bonnet-Myers theorem and the almost splitting theorem. Time permitting, I will also discuss the application of these tools to the topology of black hole horizons.