  The Kasparov groups KK_*(A, B) have a natural structure as pseudopolonais
groups. In this paper we analyze how this topology interacts with the terms of
the Universal Coefficient Theorem (UCT) and the splittings of the UCT
constructed by J. Rosenberg and the author, as well as its canonical three term
decomposition which exists under bootstrap hypotheses. We show that the various
topologies on Ext_{\Bbb Z}^1(K_*(A), K_*(B)) and other related groups mostly
coincide. Then we focus attention on the Milnor sequence and the fine structure
subgroup of KK_*(A, B).
  An important consequence of our work is that under bootstrap hypotheses the
closure of zero of KK_*(A, B) is isomorphic to the group Pext_{\Bbb
Z}^1(K_*(A), K_*(B)).
  Finally, we introduce new splitting obstructions for the Milnor and Jensen
sequences and prove that these sequences split if K_*(A) or K_*(B) is torsion
free.
