An Example of a Homotopy

Homotopy Let $$ \gamma_0(t) = (t,0), \quad \gamma_1(t) = (t,t), \quad t \in [0,1]. $$ Both paths go from $(0,0)$ to $(1,1)$. Define a homotopy $H : [0,1] \times [0,1] \to \mathbb{R}^2$ by $$ H(t,s) = (t, s t). $$ Proof that this is infact a homotopy: Initial path: $$ H(t,0) = (t,0) = \gamma_0(t). $$ Final path: $$ H(t,1) = (t,t) = \gamma_1(t). $$ Endpoints: $$ H(0,s) = (0,0), \quad H(1,s) = (1,s). $$ Thus, $H$ defines a homotopy between $\gamma_0$ and $\gamma_1$.