Open Ball in Real Number Line is Open Interval
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Theorem
Let $\struct {\R, d}$ denote the real number line $\R$ with the usual (Euclidean) metric $d$.
Let $x \in \R$ be a point in $\R$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$.
Then $\map {B_\epsilon} x$ is the open interval $\openint {x - \epsilon} {x + \epsilon}$.
Proof
Let $S = \map {B_\epsilon} x$ be an open $\epsilon$-ball at $x$.
Let $y \in \map {B_\epsilon} x$.
Then:
| \(\ds y\) | \(\in\) | \(\, \ds \map {B_\epsilon} x \, \) | \(\ds \) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \map d {y, x}\) | \(<\) | \(\, \ds \epsilon \, \) | \(\ds \) | Definition of Open $\epsilon$-Ball | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \size {y - x}\) | \(<\) | \(\, \ds \epsilon \, \) | \(\ds \) | Definition of Euclidean Metric on Real Number Line | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds -\epsilon\) | \(<\) | \(\, \ds y - x \, \) | \(\, \ds < \, \) | \(\ds \epsilon\) | Definition of Absolute Value | ||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds x - \epsilon\) | \(<\) | \(\, \ds y \, \) | \(\, \ds < \, \) | \(\ds x + \epsilon\) | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds y\) | \(\in\) | \(\, \ds \openint {x - \epsilon} {x + \epsilon} \, \) | \(\ds \) | Definition of Open Real Interval |
As the implications go both ways:
- $\map {B_\epsilon} x \subseteq \openint {x - \epsilon} {x + \epsilon}$
and
- $\map {B_\epsilon} x \supseteq \openint {x - \epsilon} {x + \epsilon}$
By definition of set equality:
- $\map {B_\epsilon} x = \openint {x - \epsilon} {x + \epsilon}$
Hence the result.
$\blacksquare$
Also see
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 $2.3.2$