Changes

::<math>H : [0,1] \times [0,1] \to X</math>

::<math>H : [0,1] \times [0,1] \to X</math>

which satisfies

which satisfies

::<math>H(t,0)=\gamma_0(t)</math>, <math>H(t,0)=\gamma_2(t)</math>, <math>H(0,s)=H(1,s)=*</math>

::<math>H(t,0)=\gamma_0(t)\,</math>, <math>H(t,0)=\gamma_2(t)\,</math>, <math>H(0,s)=H(1,s)=*\,</math>

The intuition is this: for every value of <math>s \in [0,1]</math>, we get a path <math>\gamma_s(t) : [0,1] \to X</math>, and at <math>s=0</math> and <math>s=1</math>, these are the two paths in question.  Thus <math>H</math> provides a smoothly-varying set of paths in <math>X</math>, parametrized by a variable <math>s</math>, starting with <math>\gamma_0</math>, and ending with <math>\gamma_1</math>.  This relation is often written as <math>\gamma_0 \sim \gamma_1</math>.

The intuition is this: for every value of <math>s \in [0,1]</math>, we get a path <math>\gamma_s(t) : [0,1] \to X</math>, and at <math>s=0</math> and <math>s=1</math>, these are the two paths in question.  Thus <math>H</math> provides a smoothly-varying set of paths in <math>X</math>, parametrized by a variable <math>s</math>, starting with <math>\gamma_0</math>, and ending with <math>\gamma_1</math>.  This relation is often written as <math>\gamma_0 \sim \gamma_1</math>.