Order Topology coincides with Metric Topology on the Real Numbers

Last updated Nov 1, 2022

Let $\mathbb{R}$ be viewed as a Metric Space equipped with the Euclidean Distance and as a Total Ordering with its usual Order Relation. Then the Metric Topology on $\mathbb{R}$ is the Order Topology on $\mathbb{R}$

TODO - this is plain enough to see since we can make Open Balls from Open Rays and we can take the union of Open Balls to create Open Rays.

• Real Numbers