Definition 7.3 (The fundamental group $\pi_1(X, x_0)$):

Let $X$ be a path connected topological space and $x_0$ be a point of $X$. We define $\pi_1(X, x_0)$ to be the set of all homotopy classes of paths beginning and ending at the given point $x_0$ namely homotopy classes $[\gamma]$ where $\gamma: [0,1] \rightarrow X$ is continuous and $\gamma(0) = \gamma(1) = x_0$:

$$\pi_1(X, x_0)=\{ \;[\gamma]\;/ \gamma :[0,1] \rightarrow X$$    continuous and $$\gamma(0)=\gamma(1)=x_0\}.$$