Open Ball of Metric Space/Examples/Real Number Line Example

Example of Open Ball of Metric Space
Consider the real number line with the usual (Euclidean) metric $\struct {\R, d}$.

Let $H \subseteq \R$ denote the closed real interval $\closedint 0 1$.

Let $d_H$ denote the metric induced on $H$ by $d$.

Let $\map {B_1} {1; d}$ denote the open ball of $\struct {\R, d}$ of radius $1$ and center is $1$.

Let $\map {B_1} {1; d_H}$ denote the open ball of $\struct {H, d_H}$ of radius $1$ and center is $1$.

Then by definition:
 * $\map {B_1} {1; d} = \set {x \in \R: 0 < x < 2} = \openint 0 2$

However:
 * $\map {B_1} {1; d_H} = \set {x \in \R: 0 < x \le 1} = \hointl 0 1$.