Definition:Euclidean Plus Metric

Definition

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:

$\map d {x, y} := \size {x - y} + \ds \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$

Thus $d$ adds to the Euclidean metric between $x$ and $y$ a contribution which measures the relative distances of $x$ and $y$ from $\Q$.

Also see

• Euclidean Plus Metric is Metric

Linguistic Note

The Euclidean plus metric appears not to be allocated a specific name in the literature.

The term Euclidean plus metric was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to allow it to be referred to compactly.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $30$. The Rational Numbers: $5$