Spaces modelled by an algebra on [0,∞] and their complete objects

We study constructs of type [0,∞]Set(Ω) consisting of affine sets over [0,∞] modelled by some algebra Ω. The categorical theory of closure operators is used to study separated and complete objects with respect to the Zariski closure operator, naturally defined in any category [0,∞]Set(Ω). Several basic examples are provided, in particular we show that the construct of approach spaces, the constructs of pseudo (quasi) metric spaces with contractions, the construct of topological spaces and several of its subconstructs and the construct of non-Archimedean spaces all fit into this setting.