Three-dimensional closed universes without collapse in a five-dimensional Kaluza-Klein theory

We discuss a five-vacuum and a five-dust Kaluza-Klein cosmology, in each case with the topology R×$S^3$×S. The four-space behaves in each case like a radiation-filled closed Robertson-Walker cosmology. In the five-vacuum case, the description fails at the instant of maximum expansion of the three-space. In the five-dust case, there is a genuine curvature singularity that ends the model at some nonzero value of the three-space expansion factor.