  It is conventionally believed that if a ball of matter of mass $M$ has a
radius close to $2GM$ then it must collapse to a black hole. But string theory
microstates (fuzzballs) have no horizon or singularity, and they do {\it not}
collapse. We consider two simple examples from classical gravity to illustrate
how this violation of our intuition happens. In each case the `matter' arises
from an extra compact dimension, but the topology of this extra dimension is
not trivial. The pressure and density of this matter diverge at various points,
but this is only an artifact of dimensional reduction; thus we bypass results
like Buchadahl's theorem. Such microstates give the entropy of black holes, so
these topologically nontrivial constructions dominate the state space of
quantum gravity.
