We define all paths as functions $\gamma :[0,1]\to X$ for any topological space $X$. We may also define the following:

1. Composition of paths
2. Inversion of a path
3. A closed path is one which has the same starting point as end point
4. The operation $[\alpha ][\beta ]=[\alpha \beta ]$ is well defined whenever $\alpha (1)=\beta (0)$
5. $[\alpha {]}^{-1}=$1;;$[{\alpha }^{-1}]$. The inverse in the first coordinate is not a true inverse. It just means that we only need to consider the inverse of one of the paths.
6. If $\psi :[0,1]\to [0,1]$ is a continuous bijection (reparametrization), then1;; $\alpha \simeq \alpha \circ \psi$ for any path $\alpha$.
7. Composition of paths is associative, when it is defined.
8. If $z_1\in X$ is a constant path, and $\alpha (0)=z_1$, then $[\alpha ][z_1]=[\alpha ]$.
9. Suppose $\alpha$ is any path. Then $[\alpha ][\alpha {]}^{-1}=[z_0]$ for $z_0$ in the path component of $im(\alpha )$.
10. Null Homotopy