Difference between revisions of "Homotopic paths"
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==Purpose== | ==Purpose== | ||
A homotopy is a ''continuous'' deformation from {{M|p_0}} to {{M|p_1}} | A homotopy is a ''continuous'' deformation from {{M|p_0}} to {{M|p_1}} | ||
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| + | ==Notation== | ||
| + | If {{M|p_0}} and {{M|p_1}} are end point preserving homotopic we denote this {{M|p_0\simeq p_1\text{ rel}\{0,1\} }} | ||
==See also== | ==See also== | ||
Latest revision as of 23:49, 16 April 2015
Contents
Definition
Note: by default always assume a homotopy is endpoint preserving!
Given two paths in a topological space [ilmath]p_0[/ilmath] and [ilmath]p_1[/ilmath]
Then we may say they are homotopic[1] if there exists a continuous map:
- [math]H:[0,1]\times[0,1]\rightarrow X[/math] such that
- [math]\forall t\in[0,1][/math] we have
- [math]H(t,0)=p_0(t)[/math] and
- [math]H(t,1)=p_1(t)[/math]
- [math]\forall t\in[0,1][/math] we have
End point preserving homotopy
[ilmath]H[/ilmath] is an end point preserving homotopy if in addition to the above we also have
- [math]\forall u\in[0,1]\ H(t,u)[/math] is a path from [ilmath]x_0[/ilmath] to [ilmath]x_1[/ilmath]
That is to say a homotopy where:
- [ilmath]p_0(0)=p_1(0)=x_0[/ilmath] and
- [ilmath]p_0(1)=p_1(1)=x_1[/ilmath]
Purpose
A homotopy is a continuous deformation from [ilmath]p_0[/ilmath] to [ilmath]p_1[/ilmath]
Notation
If [ilmath]p_0[/ilmath] and [ilmath]p_1[/ilmath] are end point preserving homotopic we denote this [ilmath]p_0\simeq p_1\text{ rel}\{0,1\} [/ilmath]
See also
References
- ↑ Introduction to topology - lecture notes nov 2013 - David Mond
