We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi-apartness spaces. We also deal with separation properties for spaces with inequality.

Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them.