Proof That the Real Numbers With the Topology Consisting of Open Intervals is Metrizable

Theorem

The real numbers with the topology consisting of open intervals is metrizable.

Proof

Suppose a metric spaceis given. The metricinduces a certain topologyon

In the metric spacea- ball of radiusand centreis defined as

The familyof all open balls can serve as the basis of the topology

Letbe 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.