definition of metric subspace
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 metric subspace.
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:
\( T'\): \(\in \{\text{ the metric spaces }\}\), with the metric, \(dist': T' \times T' \to \mathbb{R}\)
\(*T\): \(\subseteq T'\), \(\in \{\text{ the metric spaces }\}\), with the metric, \(dist: T \times T \to \mathbb{R}\), specified below
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Conditions:
\(\forall t_1, t_2 \in T (dist (t_1, t_2) = dist' (t_1, t_2))\)
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There is no restriction for \(T\) as a subset.
2: Note
Let us see that \(dist\) is indeed a metric for \(T\).
Let \(t_1, t_2, t_3 \in T\) be any.
\(0 \le dist (t_1, t_2) = dist' (t_1, t_2)\).
When \(t_1 = t_2\), \(dist (t_1, t_2) = dist' (t_1, t_2) = 0\).
When \(dist (t_1, t_2) = 0\), \(dist' (t_1, t_2) = dist (t_1, t_2) = 0\), so, \(t_1 = t_2\).
\(dist (t_1, t_2) = dist' (t_1, t_2) = dist' (t_2, t_1) = dist (t_2, t_1)\).
\(dist (t_1, t_3) = dist' (t_1, t_3) \le dist' (t_1, t_2) + dist' (t_2, t_3) = dist (t_1, t_2) + dist (t_2, t_3)\).
