  Huynh et al. recently showed that a countable graph $G$ which contains every
countable planar graph as a subgraph must contain arbitrarily large finite
complete graphs as topological minors, and an infinite complete graph as a
minor. We strengthen this result by showing that the same conclusion holds, if
$G$ contains every countable planar graph as a topological minor. In
particular, there is no countable planar graph containing every countable
planar graph as a topological minor, answering a question by Diestel and
K\"uhn.
  Moreover, we construct a locally finite planar graph which contains every
locally finite planar graph as a topological minor. This shows that in the
above result it is not enough to require that $G$ contains every locally finite
planar graph as a topological minor.
