Borel set: Difference between revisions
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In [[mathematics]], a '''Borel set''' is a set that belongs to the [[sigma algebra|σ-algebra]] generated by the open sets of a [[topological space]]. | In [[mathematics]], a '''Borel set''' is a set that belongs to the [[sigma algebra|σ-algebra]] generated by the open sets of a [[topological space]]. Thus, every open set is a Borel set, as are countable unions of open sets (i.e., unions of countably many open sets), and countable intersections of countable unions of open sets, etc. | ||
==Formal definition== | ==Formal definition== | ||
Let <math>(X,O)</math> be a topological space, i.e. <math>X</math> is a set and <math>O</math> are the open sets of <math>X</math> (or, equivalently, the [[topological space|topology]] of <math>X</math>). Then <math>A \subset X </math> is a Borel set of <math>X</math> if <math>A \in \sigma(O) </math>, where <math>\sigma(O)</math> denotes the σ-algebra generated by <math>O</math>. | Let <math>(X,O)</math> be a topological space, i.e. <math>X</math> is a set and <math>O</math> are the open sets of <math>X</math> (or, equivalently, the [[topological space|topology]] of <math>X</math>). Then <math>A \subset X </math> is a Borel set of <math>X</math> if <math>A \in \sigma(O) </math>, where <math>\sigma(O)</math> denotes the σ-algebra generated by <math>O</math>. | ||
The σ-algebra generated by <math>O</math> is simply the smallest σ-algebra containing the sets in <math>O</math> or, equivalently, the intersection of all σ-algebras containing <math>O</math>. | The σ-algebra generated by <math>O</math> is simply the smallest σ-algebra containing the sets in <math>O</math> or, equivalently, the intersection of all σ-algebras containing <math>O</math>.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:01, 20 July 2024
In mathematics, a Borel set is a set that belongs to the σ-algebra generated by the open sets of a topological space. Thus, every open set is a Borel set, as are countable unions of open sets (i.e., unions of countably many open sets), and countable intersections of countable unions of open sets, etc.
Formal definition
Let be a topological space, i.e. is a set and are the open sets of (or, equivalently, the topology of ). Then is a Borel set of if , where denotes the σ-algebra generated by .
The σ-algebra generated by is simply the smallest σ-algebra containing the sets in or, equivalently, the intersection of all σ-algebras containing .