  The space $D'_\Gamma$ of distributions having their wavefront sets in a
closed cone $\Gamma$ has become important in physics because of its role in the
formulation of quantum field theory in curved space time. In this paper, the
topological and bornological properties of $D'_\Gamma$ and its dual
$E'_\Lambda$ are investigated. It is found that $D'_\Gamma$ is a nuclear,
semi-reflexive and semi-Montel complete normal space of distributions. Its
strong dual $E'_\Lambda$ is a nuclear, barrelled and bornological normal space
of distributions which, however, is not even sequentially complete. Concrete
rules are given to determine whether a distribution belongs to $D'_\Gamma$,
whether a sequence converges in $D'_\Gamma$ and whether a set of distributions
is bounded in $D'_\Gamma$.
