I think this is true but i cannot prove it. Any answer or hints are welcome. 

I have tried to start with $\mathbb{R}$ with euclidean metric. We may consider $\tau :=\{\emptyset,\mathbb{R}\}$ and obvious it is a topology in $\mathbb{R}$. but in general, there are many different kinds of metric spaces so I am not sure how to prove it.