Proof That the Set of All Open Intervals is a Topology on the Set of Real Numbers
Let
be the set of all real numbers. A set
is open if for each
there exists
such that
is a metric space where
Let the family of all open intervals inbe called
then
is a topology on
Proof
Obviously
andare open inso
and
Suppose
are open sets, then
In fact
such that
where
Suppose each
is open then
is also open (
may be infinite here), since
