# PlanetPhysics/Borel Groupoid

### DefinitionsEdit

- Borel function A function ) of Borel spaces is defined to be a
*Borel function*if the inverse image of every Borel set under is also a Borel set. - Borel groupoid Let
**Failed to parse (unknown function "\grp"): {\displaystyle \grp}**be a groupoid and**Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(2)}}**a subset of**Failed to parse (unknown function "\grp"): {\displaystyle \grp \times \grp}**-- the set of its composable pairs. A*Borel groupoid*is defined as a groupoid**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B}**such that**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}}**is a Borel set in the product structure on**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B \times \grp_B}**, and also such that the functions from**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}}**to**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B}**, and from**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B}**to**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B}**are all (measurable) Borel functions (ref.^{[1]}).

#### Analytic Borel spaceEdit

**Failed to parse (unknown function "\grp"): {\displaystyle \grp_B}**
becomes an analytic groupoid if its Borel structure is analytic.

A Borel space is called *analytic* if it is
countably separated, and also if it is the image of a Borel function from a standard Borel space.

## All SourcesEdit

^{[1]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}M.R. Buneci. 2006., Groupoid C*-Algebras.,*Surveys in Mathematics and its Applications*, Volume 1, p.75 .