Proof That the Real Numbers With the Topology Consisting of Open Intervals is Metrizable
The real numbers with the topology consisting of open intervals is metrizable.
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.