  We consider the Poisson Boolean continuum percolation model in n-dimensional
hyperbolic space. In 2 dimensions we show that there are intensities for the
underlying Poisson process for which there are infinitely unbounded components
in the covered and vacant regions. In n dimensions we show that if the radius
of the balls are big enough, then there are intensities for the underlying
Poisson process for which there are infinitely many unbounded components.
