  In Radhakrishnan et al. [2020], the authors empirically show that
autoencoders trained with usual SGD methods shape out basins of attraction
around their training data. We consider network functions of width not
exceeding the input dimension and prove that in this situation basins of
attraction are bounded and their complement cannot have bounded components. Our
conditions in these results are met in several experiments of the latter work
and we thus address a question posed therein. We also show that under some more
restrictive conditions the basins of attraction are path-connected. The
tightness of the conditions in our results is demonstrated by means of several
examples. Finally, the arguments used to prove the above results allow us to
derive a root cause why scalar-valued neural network functions that fulfill our
bounded width condition are not dense in spaces of continuous functions.
