664: Section of Continuous Surjection

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definition of section of continuous surjection

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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
• 3: Note

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Starting Context

• The reader knows a definition of continuous map.
• The reader knows a definition of surjection.

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Target Context

• The reader will have a definition of section of continuous surjection.

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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:
$T_1$: $\in \{\text{ the topological spaces }\}$
$T_2$: $\in \{\text{ the topological spaces }\}$
$\pi$: $:T_1\to T_2$, $\in \{\text{ the continuous surjections }\}$
$\ast s$: $:T_2\to T_1$, $\in \{\text{ the continuous maps }\}$
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Conditions:
$\pi \circ s:T_2\to T_2=id$
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$s$ is called "section of $\pi$".

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2: Natural Language Description

For any topological spaces, $T_1,T_2$, and any continuous surjection, $\pi :T_1\to T_2$, any continuous map, $s:T_2\to T_1$, such that $\pi \circ s:T_2\to T_2$ is the identity map, $id$, is a section of $\pi$

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3: Note

$\pi$ needs to be surjective, because otherwise, there would be a $t\in T_2$ that would not be mapped under $\pi$, and then, $\pi \circ s(t)=t$ would be impossible whatever $s$ we chose, which means that $\pi \circ s=id$ would be impossible.

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References

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