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