A homogeneous space whose complement is rigid
  We construct a homogeneous subspace of $2^\omega$ whose complement is dense in $2^\omega$ and rigid. Using the same method, assuming Martin's Axiom, we also construct a countable dense homogeneous subspace of $2^\omega$ whose complement is dense in $2^\omega$ and rigid. 
Andrea Medini, Jan van Mill, Lyubomyr Zdomskyy
math.GN math.LO