View source for Homotopy

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==Definition==
A ''homotopy'' between two [[topological space|topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}}, is a [[continuous function]]:
* {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subset\mathbb{R} }})
A homotopy is ''relative to {{M|A\in\mathcal{P}(X)}}'' if {{M|F(a,t)}} is independent of {{M|t}} for all {{M|a\in A}}
==Terminology==
The family of functions {{M|\{f_t:X\rightarrow Y\ \vert\ \forall t\in[0,1],\ f_t:x\mapsto F(x,t)\} }} are called the ''stages'' of the homotopy. So we might say:
* Let {{M|f_t}} be a stage of the homotopy {{M|F}} or something similar


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==Definition==
A ''homotopy'' from the [[topological space|topological spaces]] {{Top.|X|J}} to {{Top.|Y|K}} is a [[continuous function]]{{rATHHRMS}}{{rITTGG}}:
* {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subseteq\mathbb{R} }})

For each {{M|t\in I}} we have a function:
* {{M|F_t:X\rightarrow Y}} defined by {{M|F_t:x\mapsto F(x,t)}} - these functions, the {{M|F_t}} are called the ''stages''<ref name="ATHHRMS"/> of the homotopy.
==Applications==
{{Begin Inline Theorem}}
===[[Homotopic maps]]===
{{Begin Inline Proof}}
Conditions
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
===[[Relative homotopic maps]]===
{{Begin Inline Proof}}
Conditions
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
===[[Path homotopy]]===
{{Begin Inline Proof}}
Conditions
{{End Proof}}{{End Theorem}}
==References==
<references/>
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