View source for Homotopic maps

{{Stub page|grade=A*|msg=Needs fleshing out, more references. Made a bit of a mess out of it, but I'll leave tidying up until later!}}
==Definition==
Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|f,g:X\rightarrow Y}} be ''[[continuous]]'' [[map|maps]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}.
* We say "''{{M|f}} is homotopic to {{M|g}} (relative to {{M|A}})''" if there exists a [[homotopy]] {{M|(\text{rel }A)}}<ref group="Note">Recall a [[homotopy]] (relative to {{M|A}}) is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[unit interval]]) such that:
* {{M|1=\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]}}</ref> whose {{link|initial stage|homotopy}} is {{M|f}} and whose {{link|final stage|homotopy}} is {{M|g}}.
** This is written: {{M|f\simeq g\ (\text{rel}\ A)}} 
*** or simply {{M|f\simeq g}} if {{M|1=A=\emptyset}}
** If {{M|1=A=\emptyset}} (and we write {{M|f\simeq g}}) we may say that {{M|f}} and {{M|g}} are ''freely homotopic''
* The [[homotopy]] {{M|(\text{rel }A)}} that exists if {{M|f\simeq g\ (\text{rel }A)}}, say {{M|F:X\times I\rightarrow Y}}, with {{M|1=\forall x\in X[(F(x,0)=f(x))\wedge(F(x,1)=g(x))]}} and {{M|1=\forall a\in A\forall t\in I[F(a,t)=f(a)=g(a)]}}, is called a ''homotopy of maps''
{{Begin Notebox}}
'''Explicit definition:'''<br/>{{Begin Notebox Content}}
We say {{M|f\simeq g\ (\text{rel }A)}} (or {{M|f\simeq g}} if {{M|1=A=\emptyset}}) if:
* There exists a ''[[continuous]]'' [[map]], {{M|F:X\times I\rightarrow Y}} (a ''[[homotopy]]'') such that:
*# {{M|1=\forall x\in X[F(x,0)=f(x)]}} - the {{link|initial stage|homotopy}} of the homotopy is {{M|f}}
*# {{M|1=\forall x\in X[F(x,1)=g(x)]}} - the {{link|final stage|homotopy}} of the homotopy is {{M|g}}
*# {{M|1=\forall a\in A\forall s,t\in I[F(a,s)=F(a,t)]}}<ref group="Note">Note that if {{M|1=A=\emptyset}} then this represents no condition/constraint on {{M|F}}, as are not any {{M|a\in A}} for this to be true on!</ref> - or equivalently - {{M|1=\forall a\in A\forall t\in I[F(a,t)=f(x)=g(x)]}} - the homotopy is fixed on {{M|A}}{{End Notebox Content}}{{End Notebox}}
We can use this to define a [[relation]] on continuous maps:
* If {{M|f\simeq g\ (\text{rel }A)}} then we consider {{M|f}} and {{M|g}} related and say "{{M|f}} is ''homotopic to'' {{M|g}} ({{M|\text{rel }A}})"
'''Claim: ''' this is an equivalence relation (see: [[the relation of maps being homotopic is an equivalence relation]])
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology|Homotopy Theory}}
=OLD PAGE=
==Definition==
Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|f,g:X\rightarrow Y}} be [[continuous maps]]. The [[map|maps]] {{M|f}} and {{M|h}} are said to be ''homotopic''{{rITTMJML}} if:
* there exists a [[homotopy]], {{M|H:X\times I\rightarrow Y}}, such that {{M|1=H_0=f}} and {{M|1=H_1=g}} - here {{M|1=I:=[0,1]\subset\mathbb{R} }} denotes the [[unit interval]].
*: (Recall for {{M|t\in I}} that {{M|H_t:X\rightarrow Y}} (which denotes a [[stages of a homotopy|stage of the homotopy]]) is given by {{M|H_t:x\mapsto H(x,t)}})
{{Todo|Mention free-homotopy, warn against using null (as that term is used for loops, mention relative homotopy}}
==See also==
* [[Homotopy]] - any continuous map of the form {{M|H:X\times I\rightarrow Y}}
** [[The stages of a homotopy are continuous]]
* [[Homotopy is an equivalence relation]]
* [[Path-homotopy]]
* [[Fundamental group]]
==References==
<references/>
{{Homotopy theory navbox|plain}}
{{Topology navbox}}
{{Definition|Topology|Algebraic Topology|Homotopy Theory}}[[Category:Equivalence relations]]