\begin{definition}[Definition:Fundamental Group]
Let $\struct {X, x_0}$ be a pointed topological space with base point $x_0$.
The '''fundamental group''' $\map {\pi_1} {X, x_0}$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths.
\end{definition}