Homotopic paths

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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)

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 p0p1 rel{0,1}

See also

References

  1. Introduction to topology - lecture notes nov 2013 - David Mond