definition of distance between subset and 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.
- The reader knows a definition of distance between subsets on metric space.
Target Context
- The reader will have a definition of distance between subset and 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:
\( M\): \(\in \{\text{ the metric spaces }\}\)
\( S\): \(\subseteq M\)
\( m\): \(\in M\)
\(*dist (S, m)\): \(\in \mathbb{R}\), \(= dist (S, \{m\})\)
\(*dist (m, S)\): \(\in \mathbb{R}\), \(= dist (\{m\}, S)\)
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Conditions:
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2: Note
More explicitly, \(dist (S, m) = Inf (\{dist (s, m) \vert s \in S\})\).
\(0 \le dist (S, m)\) and \(0 \le dist (m, S)\).
\(dist (S, m) = dist (m, S)\).
The triangle inequality, \(dist (S, m) \le dist (S, m') + dist (m', m)\), holds, by the proposition that for any metric space and any subset, the distance between the subset and any point satisfies the triangle inequality with respect to any other point: compare with the proposition that for any metric space, a distance between some subsets does not necessarily satisfy the triangle inequality.
The distance between points, \(dist (m, m')\), is \(dist (\{m\}, m') = dist (m, \{m'\}) = dist (\{m\}, \{m'\})\).
