definition of distance between subsets on metric space
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
- The reader knows a definition of infimum of subset of partially-ordered set.
Target Context
- The reader will have a definition of distance between subsets 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:
\( M\): \(\in \{\text{ the metric spaces }\}\)
\( S_1\): \(\subseteq M\)
\( S_2\): \(\subseteq M\)
\(*dist (S_1, S_2)\): \(\in \mathbb{R}\), \(= Inf (\{dist (s_1, s_2) \vert s_1 \in S_1, s_2 \in S_2\})\)
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Conditions:
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2: Note
\(dist (S_1, S_2)\) exists for each \(S_1\) and \(S_2\), because \(0 \le dist (s_1, s_2)\) and any lower bounded subset of \(\mathbb{R}\) has the infimum, as is well known.
\(0 \le dist (S_1, S_2)\), because \(0 \le dist (s_1, s_2)\).
\(dist (S_1, S_2) = dist (S_2, S_1)\), because \(Inf (\{dist (s_1, s_2) \vert s_1 \in S_1, s_2 \in S_2\}) = Inf (\{dist (s_2, s_1) \vert s_2 \in S_2, s_1 \in S_1\})\).
But the triangle inequality, "\(dist (S_1, S_2) \le dist (S_1, S_3) + dist (S_3, S_2)\)", does not necessarily hold, by the proposition that for any metric space, a distance between some subsets does not necessarily satisfy the triangle inequality.
