618: Deformation Retraction

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definition of deformation retraction

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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 topological subspace.
• The reader knows a definition of retraction.
• The reader knows a definition of homotopic maps.

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Target Context

• The reader will have a definition of deformation retraction.

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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:
$T^{\prime }$: $\in \{\text{ the topological spaces }\}$
$T$: $\subseteq T^{\prime }$, with the subspace topology
$\iota$: $:T\to T^{\prime },p\mapsto p$
$id$: $:T^{\prime }\to T^{\prime },p\mapsto p$
$\ast f$: $:T^{\prime }\to T$, $\in \{\text{ the retractions }\}$
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Conditions:
$\iota \circ f\simeq id$, where $\simeq$ denotes being homotopic
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2: Natural Language Description

For any topological space, $T^{\prime }$, and any subspace, $T\subseteq T^{\prime }$, any retraction, $f:T^{\prime }\to T$, such that $\iota \circ f\simeq id$, where $\iota :T\to T^{\prime },p\mapsto p$, $id:T^{\prime }\to T^{\prime },p\mapsto p$, and $\simeq$ denotes being homotopic

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References

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