Let $n \in \mathbb N$, and let $C \subset \mathbb R^n$ be convex. Equip $C$ with the subspace topology inherited from the Euclidean topology on $\mathbb R^n$. Then $C$ is path-connected: for every $x,y \in C$, there exists a continuous map $\gamma : [0,1] \to C$ such that $\gamma(0)=x$ and $\gamma(1)=y$.