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The Basic Picture: origins

The basic picture is an outcome of the development of formal topology, which emerged when looking for a precise justification of definitions. When analysed predicatively, the traditional definition of topological space leads to the underlying notion of basic pair: two sets (one of concrete points, one of observables, or indexes for basic neighbourhoods) and an arbitrary relation between them (called forcing).

A (concrete) topological space is obtained by requiring that extensions of observables form a base for a topology, which amounts to a condition on the forcing relation. So one can see that the notions of open and of closed subsets can be defined on any basic pair.

The first new discovery (reached in December '95 while looking for a predicative notion of closed subset) was that a clear logical duality is present: when closed subsets are defined primitively, they turn out to be universal-existential images of subsets under the forcing relation, while open subsets are existential-universal images.

A basic pairs can be seen as a generalized space. The symmetry between the two sets becomes a symmetry between the traditional, pointwise and the constructive, pointfree approach to topology.

Also the notion of continuity between two basic pairs acquires a purely structural explanation: it is a pair of relations such that a commutative square is produced. This is the essence of continuity.

Finally, all notions of the pointfree approach are obtained by taking the properties induced on the side of observables ( the formal side) as a defining condition.

The basic picture is the new field of mathematics which is originated from the discovery and development of such symmetries and dualities.