## COLLOQUIA PATAVINA |

Giovanni Sambin (Univ. of Padova)

Why point-free topology is more general than standard topology: an17.4.2012

embedding of concrete spaces into positive topologies

Brief
bio

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 a dynamic constructivism. In 2005, in collaboration with M. E. Maietti, he introduced a foundational theory according to such principles, called minimalist.