"new foundations must give new math"

from "The Basic Picture. Structures for constructive topology"

by Giovanni Sambin, Oxford University Press, forthcoming


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AILA (Italian Association of Logic and Applications), whose first president was Giovanni Sambin (from 1987 to 1994).

Abstracts » Valentini

Silvio Valentini

Relative Formal Topology: ten years later

Ten years ago relative formal topologies have been proposed as a way to combine a binary positivity predicate with a suitable cover relation in such a way that they enjoy both monotonicity and positivity.

Now they are a suitable tool to make explicit the relevance of the binary positivity predicate, to simplify the definition of formal point and continuous relation and to deal with the topology induced on closed subsets.

Here we show how to provide a complete formalisation for relative formal topologies both in an impredicative and a predicative setting, we propose for them an axiomatic definition and we provide a complete characterisation of continuous relations which suggests the definition of a (new?) category of relative formal topologies.