Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined

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The two main views in modern constructive mathematics usually associated with constructive type theory and topos theory are compatible with the classical view, but they are incompatible with each other, in a sense explained by some specific results which are briefly reviewed. This chapter argues in favour of a minimal foundational theory. On the one hand, this has to satisfy the proofs-as-programs paradigm, and thus be suitable for the implementation of mathematics on a computer. On the other hand, it has to be compatible with all the theories in which mathematics has been developed, like Zermelo-Fraenkel set theory, topos theory, and Martin-Löf's type theory. As a first step towards a foundational theory of this kind, the chapter formulates a specific intensional type theory, but it also warns that one should give up the expectation of an all-embracing foundation.

This paper contains: 1. a survey of what has been achieved in formal topology, 2. a short introduction to the Basic Picture, the new discipline which generalizes formal topology (on which a book by the same author is going to be published) 3. some philosophical considerations on the constructive development of topolog

We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic. quantum and non-modal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterize B positively: reflection, symmetry and visibility.A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection.To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cut-elimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.