  In teleparallel geometries the coframe and corresponding spin-connection are
the principal geometric objects and consequently the appropriate definition of
a symmetry is that of an affine symmetry. The set of invariant coframes and
their corresponding spin connections that respect the full six dimensional Lie
algebra of Robertson-Walker affine symmetries are displayed and discussed. We
will refer to such geometries as teleparallel Robertson-Walker (TRW)
geometries, where the corresponding derived metric is of Robertson-Walker form
and is characterized by the parameter $k = (-1,0,1)$. The field equations are
explicitly presented for the $F(T)$ class of teleparallel TRW spacetimes. We
are primarily interested in investigating the $k \neq 0$ TRW models. After
first studying the $k=0$ models and, in particular, writing their governing
field equations in an appropriate form, we then study their late time stability
with respect to perturbations in $k$ in both the cases of a vanishing and
non-vanishing effective cosmological constant term. As an illustration we
consider both quadratic $F(T)$ theories and power-law solutions.
