  We define enlargeable length-structures on closed topological manifolds and
then show that the connected sum of a closed $n$-manifold with an enlargeable
Riemannian length-structure with an arbitrary closed smooth manifold carries no
Riemannian metrics with positive scalar curvature. We show that closed smooth
manifolds with a locally CAT(0)-metric which is strongly equivalent to a
Riemannian metric are examples of closed manifolds with an enlargeable
Riemannian length-structure. Moreover, the result is correct in arbitrary
dimensions based on the main result of a recent paper by Schoen and Yau.
  We define the positive $MV$-scalar curvature on closed orientable topological
manifolds and show the compactly enlargeable length-structures are the
obstructions of its existence.
