Definition:Distance/Sets/Real Numbers

Definition

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

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

Also denoted as

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

Examples

Example 1

Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:

$S := \set {0, 1, 2}$

Then:

$\map d {3, S} = 1$

Example 2

Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:

$S := \openint 0 1$

Then:

$\map d {3, S} = 1$

Example 3

Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:

$S := \closedint 1 2$

Then:

$\map d {3, S} = 1$

Example 4

Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:

$S := \openint 2 3$

Then:

$\map d {3, S} = 0$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: Exercise $\S 2.13 \ (4)$