Let $(X,\tau)$ be a nonempty [topological space](/page/Topological%20Space), and let $x_0 \in X$. Then $X$ is path-connected if and only if for every $x \in X$, there exists a continuous map $\gamma : [0,1] \to X$ such that $\gamma(0)=x_0$ and $\gamma(1)=x$.