The fundamental groups of one-dimensional spaces and spatial homomorphisms

Let X be a one-dimensional metric space and ℍ be the Hawaiian earring.(1) Each homomorphism from π₁(ℍ) to π₁(X) is induced from a continuous map up to the base-point-change isomorphism on π₁(X).(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of ℍ if and only if π₁(X) is isomorphic to π₁(ℍ), if and only if X has a unique point at which X is not semi-locally simply connected. (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if π₁(X) and π₁(Y) are isomorphic. Moreover, each isomorphism from π₁(X) to π₁(Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism.