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SHAPE INJECTIVITY IS NOT IMPLIED BY BEING STRONGLY HOMOTOPICALLY HAUSDORFF

Given a topological space X and the natural homomorphism π1(X) ψ → π̌1(X) where π̌1(X) is the first shape group, we say that X satisfies the shape injectivity provided that ψ is one–to–one. A space X is (strongly) homotopically Hausdorff, if in every (free) non-zerohomotopic homotopy class there exists a minimal diameter for every path representing this class. Both properties and hence the question to investigate their mutual relationship have appeared in the theory of generalized covering spaces. The implication from the title is ruled out by constructing two spaces (the second of which will be a Peano continuum) which has the second, but not the