definition of uniformly Cauchy sequence of maps from set into 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 map.
Target Context
- The reader will have a definition of uniformly Cauchy sequence of maps from set into 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:
\( S\): \(\in \{\text{ the sets }\}\)
\( M\): \(\in \{\text{ the metric spaces }\}\)
\( J\): \(\subseteq \mathbb{N}\)
\(*s\): \(: J \to \{g: S \to M\}\)
//
Conditions:
\(\forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (\exists N \in J (\forall j_1, j_2 \in J \text{ such that } N \lt j_1, j_2 (\forall p \in S (dist (s (j_1) (p), s (j_2) (p)) \lt \epsilon))))\)
//
2: Note
The point is that \(N\) does not depend on \(p\).
\(M\) needs to be a metric space instead of a general topological space, because while the point of this concept is to guarantee the same-magnitude nearness between \(s (j_1) (p)\) and \(s (j_2) (p)\) for each \(p\), "same-magnitude"-ness does not make sense for a general topological space.
