# Definition:Borel Sigma-Algebra/Topological Space

## Definition

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 $S$.

### Borel Set

The elements of $\map \BB {S, \tau}$ are called the **Borel (measurable) sets** of $\struct {S, \tau}$.

## Also defined as

Some sources reserve the name **Borel $\sigma$-algebra** for $\map \BB {\R^n, \tau}$, where $\tau$ is the usual (Euclidean) topology.

## Also known as

The **Borel $\sigma$-algebra** is also found with the name **topological $\sigma$-algebra**, or even just **$\sigma$-algebra**.

When the set $S$ or the topology $\tau$ are clear from the context, one may encounter $\map \BB \tau$, $\map \BB S$ or even just $\BB$.

Some authors write $\BB^n$ for $\map \BB {\R^n, \tau}$.

## Also see

- Results about
**Borel $\sigma$-algebras**can be found**here**.

## Source of Name

This entry was named for Émile Borel.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.6$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 7$