466: Covering Map

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definition of 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

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Starting Context

• The reader knows a definition of connected topological space.
• The reader knows a definition of locally path-connected topological space.
• The reader knows a definition of surjection.
• The reader knows a definition of continuous map.
• The reader knows a definition of neighborhood of point.
• The reader knows a definition of homeomorphism.

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Target Context

• The reader will have a definition of 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:
$T_1$: $\in \{\text{ the connected and locally path-connected topological spaces }\}$
$T_2$: $\in \{\text{ the connected and locally path-connected topological spaces }\}$
$\ast \pi$: $:T_1\to T_2$, $\in \{\text{ the surjections }\}\cap \{\text{ the continuous maps }\}$
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Conditions:
$\mathrm{\forall }t\in T_2(\mathrm{\exists }{U}_t\subseteq T_2\in \{\text{ the open neighborhoods of }t\}\text{ such that }{U}_t\text{ is evenly-covered by }\pi )$, where "evenly-covered by $\pi$" means that for each connected component of ${\pi }^{-1}(U_t)$, ${\pi }^{-1}(U_t{)}_j$, where $j\in J$ where $J$ is a possibly uncountable index set, $\pi {|}_{{\pi }^{-1}(U_t{)}_j}:{\pi }^{-1}(U_t{)}_j\to U_t$ is a homeomorphism
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$T_2$ is called "base of covering".

$T_1$ is called "covering space of $T_2$".

Each ${\pi }^{-1}(U_t{)}_j$ is called "sheet of covering over $U_t$".

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References

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