519: Lift of Continuous Map by Covering Map

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definition of lift of continuous map by covering map

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description

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Starting Context

• The reader knows a definition of covering map.

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Target Context

• The reader will have a definition of lift of continuous map by covering map.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T_1$: $\in \{\text{ the connected topological spaces }\}\cap \{\text{ the locally path-connected topological spaces }\}$
$T_2$: $\in \{\text{ the connected topological spaces }\}\cap \{\text{ the locally path-connected topological spaces }\}$
$\pi$: $:T_1\to T_2$, $\in \{\text{ the covering maps }\}$
$T_3$: $\in \{\text{ the topological spaces }\}$
$f$: $:T_3\to T_2$, $\in \{\text{ the continuous maps }\}$
$\ast \tilde{f}$: $:T_3\to T_1$, $\in \{\text{ the continuous maps }\}$
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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 $f=\pi \circ \tilde{f}$

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References

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