# Definition:Distance/Sets

## Definition

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

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

### Metric Spaces

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

Let $x \in A$.

Let $S, T$ be subsets of $A$.

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

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

## Also denoted as

Some sources write $\operatorname{dist}$ instead of $d$.