\begin{definition}[Definition:Euclidean Plus Metric]
Let $\R$ be the set of real numbers.
Let $\set {r_i}$ be an enumeration of the rational numbers $\Q$.
The '''Euclidean plus metric''' $d: \R \times \R \to \R$ is the metric defined as:
:$\map d {x, y} := \set {x - y} + \ds \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$
Thus $d$ adds to the Euclidean metric between $x$ and $y$ a contribution which measures the relative distances of $x$ and $y$ from $\Q$.
\end{definition}