521: Maps Homotopic Relative to Subset of Domain

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definition of maps homotopic relative to subset of domain

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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 continuous map.
• The reader knows a definition of product topology.

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Target Context

• The reader will have a definition of maps homotopic relative to subset of domain.

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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:
$T_1$: $\in \{\text{ the topological spaces }\}$
$T_2$: $\in \{\text{ the topological spaces }\}$
$S$: $\subseteq T_1$
$\ast f$: $:T_1\to T_2$, $\in \{\text{ the continuous maps }\}$
$\ast f^{\prime }$: $:T_1\to T_2$, $\in \{\text{ the continuous maps }\}$
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Conditions:
$\mathrm{\exists }F:T_1\times I\to T_2,\in \{\text{ the continuous maps }\}$, where $I$ is $[0,1]\subseteq \mathbb{R}$
(
$\mathrm{\forall }p\in T_1$
(
$F(p,0)=f(p)$
$\wedge$
$F(p,1)=f^{\prime }(p)$
)
$\wedge$
$\mathrm{\forall }p\in S,\mathrm{\forall }s\in I$
(
$F(p,s)=f(p)=f^{\prime }(p)$
)
).
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$F$ is called "relative homotopy".

$f\simeq f^{\prime }relS$ denotes the relation.

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2: Natural Language Description

For any topological spaces, $T_1,T_2$, and any subset, $S\subseteq T_1$, any continuous maps, $f,f^{\prime }:T_1\to T_2$, such that there is a continuous map (called "relative homotopy"), $F:T_1\times I\to T_2$, where $I$ is $[0,1]\subseteq \mathbb{R}$, such that for each $p\in T_1$, $F(p,0)=f(p)$ and $F(p,1)=f^{\prime }(p)$, and for each $p\in S$ and each $s\in I$, $F(p,s)=f(p)=f^{\prime }(p)$, denoted as $f\simeq f^{\prime }relS$

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References

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