Making spaces wild (simply-connected case)

Abstract

We attach copies of the circle to points of a countable dense subset $D$ of a separable metric space $X$ and construct an earring space $E(X,D)$. We show that the fundamental group of $E(X,D)$ is isomorphic to a subgroup of the Hawaiian earring group, if the space $X$ is simply-connected and locally simply-connected. In addition if the space $X$ is locally path-connected, the space $X$ can be recovered from the fundamental group of $E(X,D)$.