Proof That Each Sigma Ring and Intersection of Countably Many Sets is a Borel Set
A nonempty family of sets
with
(
is the discrete topology on
) is called a
- ring if
1.
2.
If
is a topological space then a unique smallest- ring
containing the topology
exists .is called the the family of Borel sets in
Let
with each
closed inThe sets
are then open. Sinceis a- ring with
each open set
for all
Hence for all
and
If
is an intersection of countably many open sets then
where each
is open but
Eachis open therefore
