Category:Borel Sigma-Algebras

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This category contains results about Borel Sigma-Algebras.
Definitions specific to this category can be found in Definitions/Borel Sigma-Algebras.

Topological Space

Let $\struct {S, \tau}$ be a topological space

The Borel sigma-algebra $\map \BB {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$.

That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.

Metric Space

Let $\struct {S, d}$ be a metric space.

The Borel sigma-algebra (or $\sigma$-algebra) on $\struct {S, d}$ is the $\sigma$-algebra generated by the open sets in $\powerset S$.

By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on a topological space.


This category has the following 2 subcategories, out of 2 total.