Definition 7.1 (homotopy of paths):

Two paths $\gamma_0,\; \gamma_1$ in $X$ with parameter interval $[0, 1]$ such that

$\gamma_0(0) = \gamma_1(0),\; \gamma_0(1) = \gamma_1(1)$ (that is with the same end points) are said to be homotopic if there exists a continuous map $F:[0,1]\times [0,1] \rightarrow X$ such that

$$F(0,t) = \gamma_0(t)$$

$$F(1,t) = \gamma_1(t)$$

$$F(s,0) = \gamma_0(0)=\gamma_1(0)$$

$$F(s,1) = \gamma_0(1)=\gamma_1(1)$$

The definition says that the path $\gamma_0(t)$ can be continuously deformed into $\gamma_1(t)$ and $F$ is the continuous function that does the deformation. The deformation takes place in unit time parametrized by $s$. For $s\in [0,1]$, the function $\gamma_s\;:\;t\rightarrow F(s,t)$ is the intermediate path. Finally, the conditions

$$F(s,0) = \gamma_0(0)\;\;$$    and $$\;\; F(s,1)=\gamma_0(1)$$

imply that the ends $\gamma_0(0),\; \gamma_0(1)$ do not move during the deformation. in

We shall now show that homotopy is an equivalence relation.