Is it possible that in a metric space $(X, d)$ with more than one point, the only open sets are $X$ and $\emptyset$? 
Is it possible that in a metric space $(X, d)$ with more than one point, the only open sets are $X$ and $\emptyset$?

I don't think this is possible in $\mathbb{R}$, but are there any possible metric spaces where that would be true?

One of the axioms is that for $x, y \in X$ we have $d(x, y) = 0$ if and only if $x = y$. So if you have two distinct points, you should be able to find an open ball around one of them that does not contain the other.
