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,1)=\gamma_1(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>.

The constant path is defined by <math>\gamma_c(t) = *</math>, for all <math>t</math>.  A space is said to be simply-connected if every loop based at <math>*</math> is homotopic to the constant path.  This definition serves to make the intuition in terms of shrinking rubber bands precise.

The constant path is defined by <math>\gamma_c(t) = *\,</math>, for all <math>t</math>.  A space is said to be simply-connected if every loop based at <math>*</math> is homotopic to the constant path.  This definition serves to make the intuition in terms of shrinking rubber bands precise.

'''Problem:''' Prove that the relation of homotopy of paths is an [[equivalence relation]] on the set of paths.

'''Problem:''' Prove that the relation of homotopy of paths is an [[equivalence relation]] on the set of paths.