Definition:Distance/Points

Definition

Metric Space

Let $\left({X, d}\right)$ be a metric space.

The metric $d: X \times X \to \R$ is known as a distance function.

Real Numbers

Let $x, y \in \R$ be real numbers.

Let $\left|{x - y}\right|$ be the absolute value of $x - y$.

Then the function $d \left({x, y}\right) = \left|{x - y}\right|$ is called the distance between $x$ and $y$.

It is easy to show that distance as defined here is a metric.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$