Algebraic Topology/The fundamental group

Definition (fundamental group):

Let $X$ be a topological space and let $x_{0}\in X$. The fundamental group of $X$ based at $x_{0}$ is the group of homotopy equivalence classes of loops at $x_{0}$ leaving the endpoints fixed

$\pi _{1}(X,x_{0}):=\{[\gamma ]|\gamma :[0,1]\to X,\gamma (0)=\gamma (1)=x_{0}\}$

where the group law is given by concatenation of paths.