Difference between revisions of "Topological retraction/Definition"
From Maths
m |
m (Typo) |
||
| Line 1: | Line 1: | ||
<noinclude> | <noinclude> | ||
==Definition== | ==Definition== | ||
| − | </noinclude>Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be considered a s [[ | + | </noinclude>Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be considered a s [[subspace topology|subspace]] of {{M|X}}. A [[continuous map]], {{M|r:X\rightarrow A}} is called a ''retraction'' if{{rITTMJML}}: |
* The [[restriction]] of {{M|r}} to {{M|A}} (the map {{M|r\vert_A:A\rightarrow A}} given by {{M|r\vert_A:a\mapsto r(a)}}) is the [[identity map (topology)|identity map]], {{M|\text{Id}_A:A\rightarrow A}} given by {{M|\text{Id}_A:a\mapsto a}} | * The [[restriction]] of {{M|r}} to {{M|A}} (the map {{M|r\vert_A:A\rightarrow A}} given by {{M|r\vert_A:a\mapsto r(a)}}) is the [[identity map (topology)|identity map]], {{M|\text{Id}_A:A\rightarrow A}} given by {{M|\text{Id}_A:a\mapsto a}} | ||
If there is such a retraction, we say that: ''{{M|A}} is a retract<ref name="ITTMJML"/> of {{M|X}}''.<noinclude> | If there is such a retraction, we say that: ''{{M|A}} is a retract<ref name="ITTMJML"/> of {{M|X}}''.<noinclude> | ||
Revision as of 14:45, 10 May 2016
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be considered a s subspace of [ilmath]X[/ilmath]. A continuous map, [ilmath]r:X\rightarrow A[/ilmath] is called a retraction if[1]:
- The restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath] (the map [ilmath]r\vert_A:A\rightarrow A[/ilmath] given by [ilmath]r\vert_A:a\mapsto r(a)[/ilmath]) is the identity map, [ilmath]\text{Id}_A:A\rightarrow A[/ilmath] given by [ilmath]\text{Id}_A:a\mapsto a[/ilmath]
If there is such a retraction, we say that: [ilmath]A[/ilmath] is a retract[1] of [ilmath]X[/ilmath].
