Borel space

Definition 0.1.

A Borel space $(X;ℬ(X))$ is defined as a set $X$, together with a Borel $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) $ℬ(X)$ of subsets of $X$, called Borel sets. The Borel algebra on $X$ is the smallest $\sigma$-algebra containing all open sets (or, equivalently, all closed sets if the topology on closed sets is selected).

Remark 0.1.

Borel sets were named after the French mathematician Emile Borel.

Remark 0.2.

A subspace of a Borel space $(X;ℬ(X))$ is a subset $S\subset X$ endowed with the relative Borel structure, that is the $\sigma$-algebra of all subsets of $S$ of the form $S\bigcap E$, where $E$ is a Borel subset of $X$.

Definition 0.2.

A rigid Borel space $(X_r;ℬ(X_r))$ is defined as a Borel space whose only automorphism $f:X_r\to X_r$ (that is, with $f$ being a bijection, and also with $f(A)=f^{-1}(A)$ for any $A\in ℬ(X_r)$) is the identity function $1_{(X_r;ℬ(X_r))}$ (ref.[2]).

Remark 0.3.

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

Title	Borel space
Canonical name	BorelSpace
Date of creation	2013-03-22 18:23:02
Last modified on	2013-03-22 18:23:02
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	22
Author	bci1 (20947)
Entry type	Definition
Classification	msc 60A10
Classification	msc 28C15
Classification	msc 28A12
Classification	msc 54H05
Classification	msc 28A05
Synonym	measurable space
Related topic	BorelSet
Related topic	SigmaAlgebra
Related topic	MeasurableSpace
Related topic	BorelMeasure
Related topic	BorelGroupoid
Related topic	BorelMorphism
Defines	rigid Borel space
Defines	Borel subset space