773: Open Ball Around Point on Metric Space

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definition of open ball around point on metric space

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Topics

About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of metric space.

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Target Context

• The reader will have a definition of open ball around point on metric space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T$: $\in \{\text{ the metric spaces }\}$
$t$: $\in T$
$ϵ$: $\in \mathbb{R}\setminus \{0\}$
$\ast B_{t,ϵ}$: $=\{t^{\prime }\in T|dist(t,t^{\prime })<ϵ\}$
//

Conditions:
//

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2: Natural Language Description

For any metric space, $T$, any point, $t\in T$, and any $ϵ\in \mathbb{R}\setminus \{0\}$, the subset, $B_{t,ϵ}=\{t^{\prime }\in T|dist(t,t^{\prime })<ϵ\}\subseteq T$

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3: Note

The open ball does not necessarily mean that there is a point, $p\in B_{t,ϵ}$, such that $dist(t,p)=r$ for each $r<ϵ$, which is fine.

For a subspace, $T\subseteq {\mathbb{R}}^d$, with ${\mathbb{R}}^d$ regarded as the Euclidean metric space, a point, $t\in T$, and an $ϵ$, $B_{t,ϵ}$ may not be any open ball on ${\mathbb{R}}^d$, but it is an open ball on $T$ all right: an open ball on $T$ does not need to be an open ball on ${\mathbb{R}}^d$.

The open balls on the Euclidean topological space, ${\mathbb{R}}^d$, are exactly the open balls on the Euclidean metrics space, ${\mathbb{R}}^d$.

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References

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