Homotopic paths
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Contents
[hide]Definition
Note: by default always assume a homotopy is endpoint preserving!
Given two paths in a topological space p0 and p1
Then we may say they are homotopic[1] if there exists a continuous map:
- H:[0,1]×[0,1]→X such that
- ∀t∈[0,1] we have
- H(t,0)=p0(t) and
- H(t,1)=p1(t)
- ∀t∈[0,1] we have
End point preserving homotopy
H is an end point preserving homotopy if in addition to the above we also have
- ∀u∈[0,1] H(t,u) is a path from x0 to x1
That is to say a homotopy where:
- p0(0)=p1(0)=x0 and
- p0(1)=p1(1)=x1
Purpose
A homotopy is a continuous deformation from p0 to p1
Notation
If p0 and p1 are end point preserving homotopic we denote this p0≃p1 rel{0,1}
See also
References
- Jump up ↑ Introduction to topology - lecture notes nov 2013 - David Mond