  When doing representation learning on data that lives on a known non-trivial
manifold embedded in high dimensional space, it is natural to desire the
encoder to be homeomorphic when restricted to the manifold, so that it is
bijective and continuous with a continuous inverse. Using topological
arguments, we show that when the manifold is non-trivial, the encoder must be
globally discontinuous and propose a universal, albeit impractical,
construction. In addition, we derive necessary constraints which need to be
satisfied when designing manifold-specific practical encoders. These are used
to analyse candidates for a homeomorphic encoder for the manifold of 3D
rotations $SO(3)$.
