# Borel sigma-algebra

If *X* is any metric space, or more generally any topological space, the -algebra generated by the family of open sets in *X* (or, equivalently, by the family of closed sets in X) is called the **Borel -algebra** on *X* and is denoted by . Its members are called **Borel sets**. thus includes open sets, closed sets, countable intersections of open sets, countable unions of closed sets, and so forth.
^{[1]}

The Borel -algebra on is denoted by .

## Generating

is generated by each of the following:

- a. the open intervals: ,
- b. the closed intervals: ,
- c. the half-open intervals: , or ,
- d. the open rays: or ,
- e. the closed rays: or ,

## Product Borel -algebra

Let be metric spaces and let , equipped with the produc metric. Then . If the 's are separable, then .

**Proof.** By proposition in product -algebra, is generated by the sets , where is open in . Since these sets are open in , . Suppose is a countable dense set in , and let be the collection of balls in with rational radius and center in . Then every open set in is a union of members of , which is countable. Moreover, the set of points in *X* whose *j*th coordinate is in for all *j* is a countable dense subset of *X*, and the balls of radius *r* in *X* are merely products of balls of radius *r* in the 's. It follows that is generated by and is generated by . Therefore, .

**Corallary.**

- ↑ Gerald B. Folland,
*Real Analysis: Modern Techniques and Their Applications, Second Edition*, §1.2