Theorem
The real numbers with the topology consisting of open intervals is metrizable.
Proof
Suppose a metric space
is given. The metric
induces a certain topology
on
In the metric spacea- ball of radius
and centre
is defined as
The family
of all open balls can serve as the basis of the topology
Let
be the space of real numbers with the Euclidean metric. This metric induces a topology with basis consisting of open intervals. Hencewith topology generated by the basis of open intervals derived from the Euclidean metric is metrizable.