  It is known that shape injectivity implies homotopical Hausdorff and that the
converse does not hold, even if the space is required to be a Peano continuum.
This paper gives an alternative definition of homotopical Hausdorff inspired by
a new topology on the set of fixed endpoint homotopy classes of paths. This
version is equivalent to shape injectivity for Peano spaces.

Geometric Topology