  We consider the category of all locally Lipschitz contractible metric spaces
and all locally Lipschitz maps, which is a wide class of metric spaces,
including all finite dimensional Alexandrov spaces and all CAT spaces. We also
consider the chain complex of normal currents with compact support in a metric
space in the sense of Ambrosio and Kirchheim. In the present paper, its
homology is proved to be a homotopy invariant on the category.
  To prove this result, we define a new topology on a space of Lipschitz maps
between arbitrary metric spaces. This topology is proved to coincide with the
usual $C^1$-topology on the space of $C^1$-maps between compact Riemannian
manifolds.
