definition of lift of continuous map by covering map
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of covering map.
Target Context
- The reader will have a definition of lift of continuous map by covering map.
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:
\( T_1\): \(\in \{\text{ the connected topological spaces }\} \cap \{\text{ the locally path-connected topological spaces }\}\)
\( T_2\): \(\in \{\text{ the connected topological spaces }\} \cap \{\text{ the locally path-connected topological spaces }\}\)
\( \pi\): \(: T_1 \to T_2\), \(\in \{\text{ the covering maps }\}\)
\( T_3\): \(\in \{\text{ the topological spaces }\}\)
\( f\): \(: T_3 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(*\tilde{f}\): \(: T_3 \to T_1\), \(\in \{\text{ the continuous maps }\}\)
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Conditions:
\(f = \pi \circ \tilde{f}\).
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2: Natural Language Description
For any connected and locally path-connected topological spaces, \(T_1, T_2\), any covering map, \(\pi: T_1 \to T_2\), any topological space, \(T_3\), and any continuous map, \(f: T_3 \to T_2\), any continuous map, \(\tilde{f}: T_3 \to T_1\), such that \(f = \pi \circ \tilde{f}\)
