description/proof of that for metric space, bounded subset, and real number, union of open balls around each point of subset with number radius is bounded
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of bounded subset of metric space.
- The reader knows a definition of open ball around point on metric space.
Target Context
- The reader will have a description and a proof of the proposition that for any metric space, any bounded subset, and any positive real number, the union of the open balls around each point of the subset with the number radius is bounded.
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:
\(M\): \(\in \{\text{ the metric spaces }\}\)
\(S\): \(\in \{\text{ the bounded subsets of } M\}\), with diameter, \(D\)
\(\epsilon\): \(\in \mathbb{R}\), such that \(0 \lt \epsilon\)
\(S'\): \(= \cup_{s \in S} B_{s, \epsilon}\)
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Statements:
\(S'\in \{\text{ the bounded subsets of } M\}\), with diameter equal to or smaller than \(D + 2 \epsilon\)
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2: Note
This proposition does not say that the diameter of \(S'\) is exactly \(D + 2 \epsilon\).
For example, for the metric subspace of the Euclidean metric space, \(M \subseteq \mathbb{R}^2 = B_{0, 1}\), and \(S = B_{0, 1 - \epsilon / 2}\), the diameter of \(S\) is \(2 (1 - \epsilon / 2)\), but \(S' = B_{0, 1}\), whose diameter is \(2 \lt 2 (1 - \epsilon / 2) + 2 \epsilon\).
3: Proof
Whole Strategy: Step 1: take any \(s'_1, s'_2 \in S'\), and see that \(dist (s'_1, s'_2) \lt D + 2 \epsilon\).
Step 1:
Let \(s'_1, s'_2 \in S'\) be any.
\(s'_j \in B_{s_j, \epsilon}\) for an \(s_j \in S\).
\(dist (s'_1, s'_2) \le dist (s'_1, s_1) + dist (s_1, s_2) + dist (s_2, s'_2) \lt \epsilon + D + \epsilon = D + 2 \epsilon\).
So, \(S'\) is bounded with diameter equal to or smaller than \(D + 2 \epsilon\).
