definition of open ball around point on metric space
Topics
About: metric space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of metric space.
Target Context
- The reader will have a definition of open ball around point on metric space.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T\): \(\in \{ \text{ the metric spaces } \}\)
\( t\): \(\in T\)
\( \epsilon\): \(\in \mathbb{R} \setminus \{0\}\)
\(*B_{t, \epsilon}\): \(= \{t' \in T \vert dist (t, t') \lt \epsilon\}\)
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Conditions:
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2: Natural Language Description
For any metric space, \(T\), any point, \(t \in T\), and any \(\epsilon \in \mathbb{R} \setminus \{0\}\), the subset, \(B_{t, \epsilon} = \{t' \in T \vert dist (t, t') \lt \epsilon\} \subseteq T\)
3: Note
The open ball does not necessarily mean that there is a point, \(p \in B_{t, \epsilon}\), such that \(dist (t, p) = r\) for each \(r \lt \epsilon\), 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 \(\epsilon\), \(B_{t, \epsilon}\) 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, \(\mathbb{R}^d\).
