TITLE: Is there a metric space on $\mathbb{R}$ such that open sets of Euclidean metric space are not open?
QUESTION [7 upvotes]: If $X$ is metric space together with a metric function $d$, we call $U \subseteq X$ open if for all $u \in U$ there is some $\epsilon > 0$ with $B_{\epsilon}(u) \subseteq U$ where $B_{\epsilon}(u) = \{v \in X|d(u,v) < \epsilon \}$. 
So, my question is that can we have a metric on $\mathbb{R}$ such that open intervals (of Euclidean metric) are not open in this particular metric we provide? Or, can we prove that open intervals are always open in any other metric? In latter case, of course open intervals can be closed and open.

REPLY [4 votes]: Take a random, totally crazy bijection $h:\Bbb R\to \Bbb R$, then define a metric $d'$ on $\Bbb R$ by making $h$ an isometry from $(\Bbb R, d)$, with the Euclidean metric, to $(\Bbb R, d')$. Or take a bijection $\Bbb R^2\to \Bbb R$, and do basically the same thing.