  This dissertation discusses some properties of topologically ordered states
as they appear in the setting of infinite quantum spin systems. We begin by
studying the set of infinite volume ground states for Kitaev's abelian quantum
double models. We show that states describing a single excitation in the bulk
are infinite volume ground states, that is, local perturbations cannot remove
the charge. The single excitations states, which are inequivalent for distinct
charges, give a complete characterization of the sector theory for the set of
ground states. Furthermore, any pure ground state is equivalent to some single
excitation ground state. We proceed to study the stability of charges that are
classified by certain representations of the algebra of observables. We
introduce a new superselection criterion selecting almost localized and
transportable $*$-endomorphisms with respect to a vacuum state. Equivalence
classes of representations satisfying the superselection criterion form
different charged superselection sectors of the system. We show that if the
vacuum state satisfies certain locality conditions then the superselection
structure will be a braided tensor $C^*$-category.
  Further, this superselection structure is stable up to deformations by a
quasi-local dynamics.
  This result is then applied to show that the anyon structure of the abelian
quantum double models is stable under local perturbations.
