Difference between revisions of "Homotopic paths"

Latest revision as of 23:49, 16 April 2015

Definition

Note: by default always assume a homotopy is endpoint preserving!

Given two paths in a topological space $p_0$ and $p_1$

Then we may say they are homotopic^[1] if there exists a continuous map:

• $$H:[0,1]\times[0,1]\rightarrow X$$ such that
  • $$\forall t\in[0,1]$$ we have
    • $$H(t,0)=p_0(t)$$ and
    • $$H(t,1)=p_1(t)$$

End point preserving homotopy

$H$ is an end point preserving homotopy if in addition to the above we also have

• $$\forall u\in[0,1]\ H(t,u)$$ is a path from $x_0$ to $x_1$

That is to say a homotopy where:

• $p_0(0)=p_1(0)=x_0$ and
• $p_0(1)=p_1(1)=x_1$

Purpose

A homotopy is a continuous deformation from $p_0$ to $p_1$

Notation

If $p_0$ and $p_1$ are end point preserving homotopic we denote this $p_0\simeq p_1\text{ rel}\{0,1\}$

See also

• Paths and loops in a topological space

References

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