Free σ-products and fundamental groups of subspaces of the plane

Let ℍ be the so-called Hawaiian earring, i.e., ℍ = {(x,y): (x-1/n)²+y² = 1/n², 1 ≤ n ≤ ω} and o = (0,0). We prove: (1) If Y is a subspace of a line in the Euclidean plane ℝ² and X its complement ℝ²\Y with x ∈ X, then the fundamental group π₁(X, x) is isomorphic to a subgroup of π₁(ℍ, o). (2) Let Y be a subspace of a line in the Euclidean plane ℝ². Then, π₁(ℝ²\Y, x) for x ∈ ℝ²\Y is isomorphic to π₁(ℍ, o), if and only if there exists infinitely many connected components of Y which converge to a point outside of Y. (3) Every homomorphism from π₁(ℍ, o) to itself is conjugate to a homomorphism induced from a continuous map.