  In this paper we extend the theory of H-closed extensions of Hausdorff spaces
to a class of non-Hausdorff spaces, defined in \cite{B}, called $n$-Hausdorff
spaces. The notion of H-closed is generalized to an $n$-H-closed space. Known
construction for Hausdorff spaces $X$, such as the Kat\v{e}tov H-closed
extension $\kappa X$, are generalized to a maximal $n$-H-closed extension
denoted by $n$-$\kappa X$.
