Changes

An equivalence relation must also satisfy reflexivity, so that <math>\gamma_0 \sim \gamma_1</math> implies that <math>\gamma_1 \sim \gamma_0</math>.  To prove that homotopy of loops satisfy this, suppose that <math>\gamma_0 \sim \gamma_1</math>.  Then there is a homotopy <math>H_0(t,s) : [0,1] \times [0,1] \to X</math> satisfying the requisite conditions.  A homotopy from <math>\gamma_1</math> to <math>\gamma_0</math> is provided by running <math>H_0</math> in reverse: set <math>H(t,s) = H_0(t,1-s)</math>.

An equivalence relation must also satisfy reflexivity, so that <math>\gamma_0 \sim \gamma_1</math> implies that <math>\gamma_1 \sim \gamma_0</math>.  To prove that homotopy of loops satisfy this, suppose that <math>\gamma_0 \sim \gamma_1</math>.  Then there is a homotopy <math>H_0(t,s) : [0,1] \times [0,1] \to X</math> satisfying the requisite conditions.  A homotopy from <math>\gamma_1</math> to <math>\gamma_0</math> is provided by running <math>H_0</math> in reverse: set <math>H(t,s) = H_0(t,1-s)</math>.

The final requirement is that of transitivity, that <math>\gamma_0 \sim \gamma_1</math> and <math>\gamma_1 \sim \gamma_2</math> implies that <math>\gamma_0 \sim \gamma_2</math>.  This is the intuitive fact that if one path may be deformed to another, and that one may be deformed some third path, then the original path may be deformed directly to the third path.  To prove this, let <math>H_0</math> be a homotopy from <math>\gamma_0</math> to <math>\gamma_1</math>, and <math>H_1</math> a homotopy from <math>\gamma_1</math> to <math>\gamma_2</math>.  Define a new homotopy by  

The final requirement is that of transitivity, that <math>\gamma_0 \sim \gamma_1</math> and <math>\gamma_1 \sim \gamma_2</math> implies that <math>\gamma_0 \sim \gamma_2</math>.  This is the intuitive fact that if one path may be deformed to another, and that one may be deformed some third path, then the original path may be deformed directly to the third path.  To prove this, let <math>H_0</math> be a homotopy from <math>\gamma_0</math> to <math>\gamma_1</math>, and <math>H_1</math> a homotopy from <math>\gamma_1</math> to <math>\gamma_2</math>.  Define a new homotopy by  

:: <math>H(t,s) = \begin{cases} H_1(t,2s) & \textrm{if $0 \leq s \leq 1/2$,} \\ H_2(t,2s-1) & \textrm{if $1/2 \leq s \leq 1$} \end{cases} </math>.

::<math>H(t,s) =

\begin{cases}

  H_1(t,2s) & \mbox{if } 0 \leq s \leq 1/2, \\

  H_2(t,2s-1) & \mbox{if } 1/2 \leq s \leq 1

\end{cases}</math>.

This is a homotopy from <math>\gamma_0</math> to <math>\gamma_2</math>.

This is a homotopy from <math>\gamma_0</math> to <math>\gamma_2</math>.