Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), and let $x,y,z \in X$. Let $\gamma : [0,1] \to X$ be a path from $x$ to $y$, and let $\eta : [0,1] \to X$ be a path from $y$ to $z$, where $[0,1]$ has the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{R}$. Define the concatenation $\gamma * \eta : [0,1] \to X$ by the following two prescriptions: for every $t \in [0,1]$ satisfying $0 \leq t \leq \frac{1}{2}$,

\begin{align*} (\gamma * \eta)(t)=\gamma(2t), \end{align*}

and for every $t \in [0,1]$ satisfying $\frac{1}{2} \leq t \leq 1$,

\begin{align*} (\gamma * \eta)(t)=\eta(2t-1). \end{align*}

Then $\gamma * \eta$ is a path from $x$ to $z$.