One-dimensional Peano continua with zero-dimensional wild part
Let $X^w$ be the subspace of a space $X$ consisting of all points at which $X$ is not semi-locally simply connected. Let $X$ be a one-dimensional Peano continuum. It is known that if $X^w$ is one point, then $X$ is homotopy equivalent to the Hawaiian earring. However, the homeomorphism type of $X^w$ does not determine the homotopy type of $X$ in general. Here, we show that, for a one-dimensional Peano continuum $X$ such that $X^w$ is zero-dimensional and non-empty, the homeomorphism type of $X^w$ determines the homotopy type of $X$.