  Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between
commutative $C^*$-algebras and locally compact Hausdorff spaces. So any
noncommutative $C^*$-algebra can be regarded as a generalization of a
topological space. Similarly a spectral triple is a generalization of a
Riemannian manifold. An (infinitely listed) covering of a Riemannian manifold
has natural structure of Riemannian manifold. Here we will consider the
noncommutative generalization of this result.
