Based on Connes' notion of an unbounded Fredholm module for a C*-algebra and the work by Rieffel on metrics on state spaces for C*-algebras we have searched for, and found a spectral triple for a UHF C*-algebra such that the topology it generates on the state space is the weak*-topology. We have a similar result for the compacts. We have also translated the Arzela-Ascoli Theorem to the language of unital non commutative C*-algebras.