A new version of homotopical Hausdorff (Q2850657)

A space \(X\) is homotopically Hausdorff if for every \(x\in X\) and for every essential loop \(\alpha\) based at \(x\), there is a neighborhood \(U\) of \(x\) such that \(\alpha\) is not homotopic to any loop in \(U\). A space \(X\) is shape injective if the naturally induced homomorphism \(\pi_1(X)\to \check \pi_1 (X)\) into the first shape group is an injection. Both separation properties of homotopy classes of loops have a role in the study of locally complicated spaces. They are closely related to the properties of path spaces and generalized covering spaces. It is well known that shape injectivity implies the property of being homotopically Hausdorff. The converse does not hold, which is demonstrated by examples in [\textit{G. Conner} et al., Topology Appl. 155, No. 10, 1089--1097 (2008; Zbl 1148.57030)].NEWLINENEWLINEIn this paper the author modifies the first definition in the following way: a space \(X\) is lasso homotopically Hausdorff if for each \(x \in X\) and each essential loop \(\gamma\) based at \(x\), there is a normal open cover \(\mathcal U\) of \(X\) such that \([\gamma ] \notin \pi_1(\mathcal U, x)\). He then proves that this condition is equivalent to shape injectivity for path connected, locally path connected spaces. The proof uses treatment of loops in terms of chains. The author also proves and uses the fact that the first shape group can be obtained using chains reflecting the structure of covers, rather that ordinary loops.