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 − 1n)2 + y2 = 1n2, 1 ⩽ n < ω} and o = (0,0). We prove: 1. Y is a subspace of a line in the Euclidean plane R2 and X its complement R2β Y with x ϵ X, then the fundamental group π1(X, x) is isomorphic to a subgroup of π1(H, o). t Y be a subspace of a line in the Euclidean plane R2. Then, π1(R2 β Y, x) for x ϵ R2 β 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. ery homomorphism from π1(H, o) to itself is conjugate to a homomorphism induced from a continuous map.