Definition:Distance

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.

Metric Spaces

Let $V$ be a metric space with associated metric $d$.

Let $x \in V$, and let $S, T$ be subsets of $V$.

The distance between $x$ and $S$ is defined and annotated $\displaystyle d \left({x, S}\right) = \inf_{y \in S} \left({d \left({x, y}\right)}\right)$.

The distance between $S$ and $T$ is defined and annotated $\displaystyle d \left({S, T}\right) = \inf_{\substack{x \in S \\ y \in T}} \left({d \left({x, y}\right)}\right)$.

Real Numbers

Let $S, T$ be a subsets of the set of real numbers $\R$.

Let $x \in \R$ be a real number.

The distance between $x$ and $S$ is defined and annotated $\displaystyle d \left({x, S}\right) = \inf_{y \in S} \left({d \left({x, y}\right)}\right)$, where $d \left({x, y}\right)$ is the distance between $x$ and $y$.

The distance between $S$ and $T$ is defined and annotated $\displaystyle d \left({S, T}\right) = \inf_{\substack{x \in S \\ y \in T}} \left({d \left({x, y}\right)}\right)$.