Open Ball of Metric Space/Examples/Real Number Line Example
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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:
- $\map {B_1} {1; d_H} = \set {x \in \R: 0 < x \le 1} = \hointl 0 1$.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.5$