View source for Topological retraction

{{Refactor notice|grade=A*|msg=Messy as ... - it's just awful}}
==Proof==
Note that if {{M|r\circ i_A\eq \text{Id}_A}} then {{M|r_*\circ(i_A)_*\eq (\text{Id}_A)_*}} 
* So {{M|r_*\circ(i_A)_*}} must be a [[bijection]]
** By [[if the composition of two functions is a bijection then the initial map is injective and the latter map is surjective]]
*** We see:
***# {{M|r_*:\pi_1(X,a)\rightarrow\pi_1(A,a)}} is [[surjective]]
***# {{M|(i_A)_*:\pi_1(A,a)\rightarrow\pi_1(X,a)}} is [[injective]]
{{Green highlight|1='''Alec's thought: ''' can we use the [[first group isomorphism theorem]] on {{M|r_*}} to get {{M|\pi_1(A,a)}} from {{M|\pi_1(X,a)}} or something?}}
=OLD PAGE=
{{Stub page|grade=A*|msg=Demote to grade A once tidied up. Find other sources. Be sure to link to [[deformation retraction]] and [[strong deformation retraction]]}}
==[[/Definition|Definition]]==
{{/Definition}}<br/>
'''Claim 1:'''
* This is equivalent to the condition: {{M|1=r\circ i_A=\text{Id}_A}} where {{M|i_A}} denotes the [[inclusion map (topology)|inclusion map]], {{M|i_A:A\hookrightarrow X}} given by {{M|i_A:a\mapsto x}}
{{Todo|In the case of {{M|1=A=\emptyset}} - does it matter? I don't think so, but check there is nothing ''noteworthy'' about it. Also proof of claims}}
==See also==
* [[Types of topological retractions]] - comparing ''retraction'' with [[deformation retraction]] and [[strong deformation retraction]]
===Important theorems===
* [[For a retraction the induced homomorphism on the fundamental group is surjective]]
** {{M|\forall p\in A}} the [[induced homomorphism on fundamental groups]] of the retraction, {{M|r_*:\pi_1(X,p)\rightarrow\pi_1(A,p)}} is [[surjective]]
* [[For the inclusion map of a retract of a space the induced homomorphism on the fundamental group is injective]]
** {{M|\forall p\in A}} the induced homomorphism on fundamental groups of the [[inclusion map (topology)|inclusion map, {{M|i_A:A\hookrightarrow X}}]], which is {{M|(i_A)_*:\pi_1(A,p)\rightarrow \pi_1(X,p)}} is [[injective]]
===Lesser theorems===
* [[A retract of a connected space is connected]]
* [[A retract of a compact space is compact]]
* [[A retract of a retract of X is a retract of X]]
* [[A retract of a simply connected space is simply connected]]
==References==
<references/>
{{Definition|Algebraic Topology|Homotopy Theory|Topology|Topological Manifolds|Manifolds}}