A new version of homotopical Hausdorff

TL;DR

This paper introduces an alternative definition of homotopical Hausdorff, showing it is equivalent to shape injectivity in Peano spaces through a new topology on path classes.

Contribution

It proposes a new topology-based definition of homotopical Hausdorff that aligns with shape injectivity in Peano spaces, clarifying their relationship.

Findings

New topology on fixed endpoint homotopy classes of paths

Equivalence of the new homotopical Hausdorff definition and shape injectivity for Peano spaces

Clarification of the relationship between shape injectivity and homotopical Hausdorff

Abstract

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.