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Some points in formal topology. (English) Zbl 1044.54001

It is now more than fifteen years since the appearance of the author’s ‘first communication’ on formal spaces, a constructive and predicative approach to general topology designed to be interpretable in Martin-Löf’s type theory.
The present paper has three main aims, which are developed in three broadly independent sections: first, to survey the progress that has been made since then in developing traditional notions of topology in the formal setting; secondly, to provide publicity for what the author calls ‘the basic picture’, which is a newer and more symmetric way of looking at the relationship between points and open sets, still under development by the author and S. Gebellato; and finally, in a section less mathematical and more philosophical in character, to present the author’s understanding of how and why one should seek to develop mathematics in a constructive context.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
03A05 Philosophical and critical aspects of logic and foundations
03F65 Other constructive mathematics
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