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

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

### Metric Spaces

Let $M = \struct {A, d}$ 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 $\ds \map d {x, S} = \inf_{y \mathop \in S} \paren {\map d {x, y} }$.

The **distance between $S$ and $T$** is defined and annotated $\ds \map d {S, T} = \inf_{\substack {x \mathop \in S \\ y \mathop \in T} } \paren {\map d {x, y} }$.

## Also denoted as

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