definition of closed ball around point on metric space
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
Target Context
- The reader will have a definition of closed 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:
\( M\): \(\in \{ \text{ the metric spaces } \}\)
\( m\): \(\in M\)
\( \epsilon\): \(\in \mathbb{R}\), such that \(0 \lt \epsilon\)
\(*B'_{m, \epsilon}\): \(= \{m' \in M \vert dist (m, m') \le \epsilon\}\)
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Conditions:
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2: Note
The closed ball does not necessarily mean that there is a point, \(p \in B'_{m, \epsilon}\), such that \(dist (m, p) = r\) for each \(r \le \epsilon\), which is fine.
For a subspace, \(M \subseteq \mathbb{R}^d\), with \(\mathbb{R}^d\) regarded as the Euclidean metric space, a point, \(m \in M\), and an \(\epsilon\), \(B'_{m, \epsilon}\) may not be any closed ball on \(\mathbb{R}^d\), but it is an closed ball on \(M\) all right: a closed ball on \(M\) does not need to be an closed ball on \(\mathbb{R}^d\).
The closed balls on the Euclidean topological space, \(\mathbb{R}^d\), are exactly the closed balls on the Euclidean metric space, \(\mathbb{R}^d\).
For \(M\) with the topology induced by the metric, \(B'_{m, \epsilon}\) is not necessarily the closure of \(B_{m, \epsilon}\), by the proposition that for any metric space with the induced topology, any closed ball is closed and contains but not necessarily equal the closure of the open ball with the same radius.
