Colloquia Patavina: Why point-free topology is more general than standard topology: an embedding of concrete spaces into positive topologies

Martedi' 17 Aprile 2012 - Giovanni Sambin


Martedi' 17 Aprile 2012 alle ore 16:00 in aula 1A150 della Torre Archimede il Professor Giovanni Sambin (Univ. Padova) terra' una conferenza della serie Colloquia Patavina.

La Commissione Colloquia
P. Ciatti, M.A. Garuti, M. Pavon, F. Rossi

Why point-free topology is more general than standard topology: an embedding of concrete spaces into positive topologies

Giovanni Sambin (Univ. Padova)

The aim of this talk is to offer a panoramic view on some recent developments in constructive topology, stressing in particular on those structures and insights which are new also for a classical mathematician. One of the motivations for point-free topology is that many properties of topological spaces depend only on the algebraic structure of open subsets, called a locale. The fundamental link with standard topology (with points) is given by a categorical adjunction between topological spaces and locales. We show that this adjunction can be refined into an embedding, provided that topological spaces are replaced with "concrete spaces" and locales with "positive topologies". We thus can give a mathematical expression to the original and deeper motivation for point-free topology, namely the claim that it is is more general than topology with points. The notion of a concrete space is reached from that of a topological space by always requiring to have a basis indexed by a set, called the set of observables. Then it is natural to pass from continuous functions to continuous relations (set-valued mappings) with a suitable equality. One thus obtains a structural characterization of continuity, namely as commutativity of a square of relations. The link between points and observables shows that interior and closure of a subset are defined by logically dual formulas, which only classically reduce to complementation. Positive topologies are the abstraction of the structure induced by a concrete space on the set of observables. Contrary to locales, they include a primitive pointfree treatment of closed subsets. All these technical novelties have been induced by the constructive approach. The systematic presence of bases is necessary in a predicative foundation (no powerset axiom) and the distinction between closed and complement of open is due to intuitionistic logic (no law of excluded middle). Constructively, the point-free approach becomes necessary, rather than just more general, since in most examples of spaces the collection of points is not a set predicatively. Thus spaces are obtained constructively as the collection of ideal points, that is, suitable subsets of observables in a positive topology.

-Breve curriculum
Born in 1948, laurea in mathematics in 1971, professor since 1981, first in Siena, later (since 1987) in Padua. Founder and first president (1987-93) of the Italian Association of Logic and Applications (AILA). Scientific coordinator since 1999 of five projects financed by the italian government on Constructive Methods in mathematics and computer science. Coming from the school of mathematical logic of R. Magari in Siena, he has been one of the founders of the modal logic of provability (the fixed point theorem by de Jongh-Sambin, 1976). In 1980-82 he took care of exposition for the only textbook by Martin-Löf on his constructive type theory. In 1984 he started formal topology (that is, topology as developed on the base of type theory), which is by now widely known as one of the most fertile branches of constructive mathematics. He has organized three Workshops on Formal Topology (1997, 2002 and 2007). In 1995 he discovered a logical duality between the notions of open and closed in topology. This brought to positive topology, a generalization of topology in which closed and open subsets are treated on the same level. In particular, he introduced a primitive notion of closed subset in pointfree topology, which starts an abstract mathematical treatment of existential statements. His book on positive topology will be published by Oxford U.P. In 1995-96 he isolated basic logic, in which logical constants are obtained through a reflection of metalanguage into object language and which gives a conceptual and geometrical structure to the space of logics. He has always cultivated active interest in foundations of mathematics. Since 1999 he has developed arguments in favour of ad ynamic constructivism. In 2005, in collaboration with M. E. Maietti, he introduced a foundational theory according to such principles, called minimalist.

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