Topological properties of non-Archimedean approach spaces

In this paper we give an isomorphic description of the category of non-Archimedian approach spaces as a category of lax algebras for the ultrafilter monad and an appropriate quantale. Non-Archimedean approach spaces are characterised as those approach spaces having a tower consisting of topologies. We study topological properties p, for p compactness and Hausdorff separation along with low-separation properties, regularity, normality and extremal disconnectedness and link these properties to the condition that all or some of the level topologies in the tower have p. A compactification technique is developed based on Shanin's method.