The closure and interior of $(0,1)$ on $\mathbb{R}$ under the Zariski topology.

On $\mathbb{R}$ under the Zariski topology I don't understand why the closure of $(0,1)$ is $\mathbb{R}$? Clearly this is the only closed set containing $(0,1)$ but I thought we're looking for an open set containing $(0,1)$ under the definition of closure.

Also, I don't understand why the interior of $(0,1)$ is $\emptyset$. Under the definition of the Zariski topology surely $(0,1)$ is open so the interior would be itself?