## Proof That Each Sigma Ring and Intersection of Countably Many Sets is a Borel Set

A nonempty family of setswith(is the discrete topology on) is called a- ring if

1.

2.

Ifis a topological space then a unique smallest- ringcontaining the topology exists .is called the the family of Borel sets in

Letwith eachclosed inThe setsare then open. Sinceis a- ring witheach open setfor all

Hence for alland

Ifis an intersection of countably many open sets then where eachis open but

Eachis open therefore