A description/proof of that for metric space, distance between points in 2 open balls is larger than distance between centers minus sum of radii and smaller than distance between centers plus sum of radii
Topics
About: topological 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 description and a proof of the proposition that for any metric space, the distance between any 2 points each of which is in its own any open ball is larger than the distance between the centers of the open balls minus the sum of the radii of the open balls and smaller than the distance between the centers of the open balls plus the sum of the radii of the open balls.
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: Description
For any metric space, \(T\), any 2 points, \(p_1, p_2 \in T\), any open balls, \(B_{p_i-\epsilon_i}\), around \(p_i\), and any 2 points, \(p'_i \in B_{p_i-\epsilon_i}\), the distance, \(dist (p'_1, p'_2)\), satisfies \(dist (p_1, p_2) - (\epsilon_1 + \epsilon_2) \lt dist (p'_1, p'_2) \lt dist (p_1, p_2) + (\epsilon_1 + \epsilon_2)\).
2: Proof
\(dist (p'_1, p_2) \leq dist (p'_1, p'_2) + dist (p'_2, p_2)\); \(dist (p'_1, p_2) - dist (p'_2, p_2) \leq dist (p'_1, p'_2)\). \(dist (p_1, p_2) \leq dist (p_1, p'_1) + dist (p'_1, p_2)\); \(dist (p_1, p_2) - dist (p_1, p'_1) \leq dist (p'_1, p_2)\). Adding the 2 results, \(dist (p_1, p_2) - dist (p_1, p'_1) - dist (p'_2, p_2) \leq dist (p'_1, p'_2)\); \(dist (p_1, p_2) - \epsilon_1 - \epsilon_2 \lt dist (p'_1, p'_2)\).
\(dist (p'_1, p'_2) \leq dist (p'_1, p_2) + dist (p_2, p'_2)\). \(dist (p'_1, p_2) \leq dist (p'_1, p_1) + dist (p_1, p_2)\). Adding the 2 results, \(dist (p'_1, p'_2) \leq dist (p_1, p_2) + dist (p'_1, p_1) + dist (p_2, p'_2)\). \(dist (p'_1, p'_2) \lt dist (p_1, p_2) + \epsilon_1 + \epsilon_2\)
3: Note
While it seems obvious for any Euclid metric space, it is true for any metric space.
