  We show that every homomorphism from a one-dimensional Peano continuum to a
planar Peano continuum is induced by a continuous map up to conjugation. We
then prove that the topological structure of the space of points at which a
planar Peano continuum is not semi-locally simply connected is determined
solely by algebraic information contained in its fundamental group.
Furthermore, we demonstrate how to reconstruct the topological structure of the
space of points at which a planar Peano continuum is not semi-locally simply
connected using only the subgroup lattice of its fundamental group.
