Is there any open set in $(\mathbb R,d)$ which is not open in $(\mathbb R,d')$ where $\mathbb R$ is set of real number?

Question

Here $(\mathbb{R},d)$ is metric space with usual distance metric and $(\mathbb{R},d')$ is metric with discrete metric. I think there is no this type of set in $(\mathbb{R},d)$ which is not open in $(\mathbb{R},d')$.

Answer 1

Every singleton set is open in $(\mathbb{R},d^\prime)$, the discrete metric space. That is because the balls of radii $< 1$ are all singletons. Since every open set in $(\mathbb{R},d)$ can be expressed as a union of singleton sets, then it is an open set in $(\mathbb{R},d^\prime)$.