  The study of cosmological perturbation theory in $f(T)$ gravity is a topic of
great interest in teleparallel gravity since this is one of the simplest
generalizations of the theory that modifies the teleparallel equivalent of
general relativity. In this work, we explore the possibility of a non-flat FLRW
background solution and perform perturbations for positively as well as
negatively curved spatial geometries, together with a comparison to the flat
case. We determine the generalized behaviour of the perturbative modes for this
non-flat FLRW setting for arbitrary $f(T)$ models, when the most general
homogeneous and isotropic background tetrads are used. We also identify
propagating modes in this setup, and relate this with the case of a flat
cosmology.
