On the Meaning of Positivity Relations for Regular Formal Spaces

A careful analysis of the original definition of formal topology led to the introduction of a new primitive, namely a positivity relation between elements and subsets. This is, in other terms, a direct intuitionistic treatment of the notion of closed subset in formal topology. However, since formal open subsets do not determine formal closed subsets uniquely, the new concept of positivity relation is not yet completely clear. Here we begin to illustrate the general idea that positivity relations can be regarded as a further, powerful tool to describe properties of the associated formal space. Our main result is that, keeping the formal cover fixed, by suitably redefining the positivity relation of a regular formal topology one can obtain any given set-indexed family of points as the corresponding formal space.