PlanetPhysics/Borel Space

A Borel space is defined as a set , together with a Borel -algebra of subsets of , called Borel sets. The Borel algebra on is the smallest -algebra containing all open sets (or, equivalently, all closed sets if the topology on closed sets is selected).

Borel sets were named after the French mathematician Emile Borel.

A subspace of a Borel space is a subset endowed with the relative Borel structure, that is the -algebra of all subsets of of the form , where is a Borel subset of .

A rigid Borel space is defined as a Borel space whose only automorphism (that is, with being a bijection, and also with for any ) is the identity function (ref.[1]).

R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a `set of large cardinality'.

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References edit

  1. 1.0 1.1 B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS. , 113 (4):1013-1015., available online.
  2. M.R. Buneci. 2006., Groupoid C*-Algebras., Surveys in Mathematics and its Applications , Volume 1: 71--98.
  3. A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in Math.}, Springer-Verlag, Berlin, {\mathbf 725}: 19-14.