Terminology:

Paths in $X$ starting and ending at $x_0$ will be referred to as loops based at $x_0$. The distinguished point $x_0 \in X$ is called the base point of $X$.

Note that if $\gamma_1,\gamma_2$ are two loops based at $x_0$, their juxtaposition $\gamma_1 \ast \gamma_2$ is defined whereby both the products $[\gamma_1][\gamma_2]$ and $[\gamma_2][\gamma_1]$ are defined. Also for $[\gamma] \in \pi_1(X,x_0)$, $[\gamma^{-1}]$ also belongs to $\pi_1(X, x_0)$. $[\varepsilon_{x_0}] \in \pi_1(X,x_0)$ and lemma (7.5) and theorem (7.6) imply that $\pi_1(X, x_0)$ is a group with unit element $[\varepsilon_{x_0}]$. This group is written multiplicatively and the unit element $[\varepsilon_{x_0}]$ will be denoted by $1$ when there is no danger of confusion. Summarizing,