definition of uniformly convergent sequence of maps from topological space into metric space
Topics
About: topological space
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- 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 convergent sequence of maps from topological space 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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the metric spaces }\}\)
\( J\): \(\subseteq \mathbb{N}\)
\(*s\): \(: J \to \{g: T_1 \to T_2\}\)
//
Conditions:
\(\exists f: T_1 \to T_2, \forall t \in T_1 (: J \to T_2, j \mapsto s (j) (t) \text{ converges to } f (t))\)
\(\land\)
\(\forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (\exists N \in \mathbb{N} (\forall n \in \mathbb{N} \text{ such that } N \lt n (\forall t \in T_1 (dist (s (j) (t), f (t)) \lt \epsilon))))\)
//
2: Note
The point is that \(N\) does not depend on \(t\).
\(T_2\) needs to be a metric space instead of a general topological space, because while the point of this concept is to take the same-sized open ball around each \(f (t)\), "same-sized"-ness does not make sense for a general topological space.
