"** new foundations must give new math**"

from

by Giovanni Sambin, Oxford University Press, forthcoming

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Abstracts »
Valentini

**Silvio Valentini**

Padova

**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.