Difference between revisions of "Notes:Homotopy terminology"
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+ | ==Plan== | ||
+ | # Homotopy - a thing, happens to be a relation on its terminal stages | ||
+ | # Homotopic - redirect to "Homotopic maps" - an equivalence relation on maps | ||
+ | # Homotopic paths - special case of homotopic maps | ||
+ | =OLD PAGE= | ||
==Homotopy== | ==Homotopy== | ||
Homotopy is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subseteq\mathbb{R} }}{{rATHHRMS}}.<br/> | Homotopy is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subseteq\mathbb{R} }}{{rATHHRMS}}.<br/> |
Revision as of 12:46, 14 September 2016
Contents
[hide]Plan
- Homotopy - a thing, happens to be a relation on its terminal stages
- Homotopic - redirect to "Homotopic maps" - an equivalence relation on maps
- Homotopic paths - special case of homotopic maps
OLD PAGE
Homotopy
Homotopy is a continuous map, F:X×I→Y where I denotes the unit interval, [0,1]⊆R[1].
Here X and Y are topological spaces
Homotopic maps
If f,g:X→Y are continuous maps, we say that "f is homotopic to g" if[2]:
- There is a homotopy, F:X×I→Y such that F(x,0)=f(x) and F(x,1)=g(x) (∀x∈X)
Homotopic relative to A
If f,g:X→Y are continuous maps and A∈P(X), we say that "f is homotopic to g relative to A" if[3]:
- There is a homotopy, F:X×I→Y such that F(x,0)=f(x) and F(x,1)=g(x) AND
- F(a,t)=f(a)=g(a) for all t∈I and ∀a∈A
Homotopic paths
This is a special case, here we are dealing with A:={0,1} and F:I×I→X, the maps we are building a homotopy between are of the form:
- α:I→X
And we say:
- α1 is homotopic to α2 (rel {0,1}) if there is a homotopy between them.