definition of bounded subset of 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 bounded subset of 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:
\( M\): \(\in \{\text{ the metric spaces }\}\)
\(*S\): \(\subseteq M\)
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Conditions:
\(\exists r \in \mathbb{R} (\forall s_1, s_2 \in S (dist (s_1, s_2) \lt r))\)
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2: Note
The infimum of the set of such \(r\) s is called "diameter of \(S\)".
For the diameter of \(S\), \(D\), \(\forall s_1, s_2 \in S (dist (s_1, s_2) \le D)\), because if \(D \lt dist (s_1, s_2)\), \(D + (dist (s_1, s_2) - D) / 2 \lt dist (s_1, s_2)\), so, all the \(r\) s would need to satisfy \(D + (dist (s_1, s_2) - D) / 2 \lt r\), which would mean that \(D + (dist (s_1, s_2) - D) / 2\) was a lower bound of all the \(r\) s and \(D\) was not the maximum of the lower bounds, a contradiction.
\(\forall s_1, s_2 \in S (dist (s_1, s_2) \lt D)\) does not necessarily hold: for example, \(M = \mathbb{R}\) as the Euclidean metric space and \(S = [-1, 1]\), then, \(D = 2\), but \(dist (-1, 1) = 2 = D\).
When \(S\) is bounded, for each fixed \(m \in M\), there is an \(r' \in \mathbb{R}\) such that for each \(s \in S\), \(dist (m, s) \lt r'\), because taking any fixed \(s' \in S\), \(dist (m, s) \le dist (m, s') + dist (s', s) \lt dist (m, s') + r\), so, \(r' := dist (m, s') + r\) will do.
On the other hand, when there is an \(r' \in \mathbb{R}\) such that for an \(m \in M\), for each \(s \in S\), \(dist (m, s) \lt r'\), \(S\) is bounded, because for each \(s_1, s_2 \in S\), \(dist (s_1, s_2) \le dist (s_1, m) + dist (m, s_2) \lt 2 r'\), so, \(r := 2 r'\) will do.
