Homotopic paths

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 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]

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

• Paths and loops in a topological space