Past exam of the metric spaces and complex analysis course of the University of Oxford / 2015 / 2 / a / ii / Solution

A metric space is path-connected if every pair $x,y$ can be joined by a path: a continuous map $\gamma :[0,1]\to X$ with $\gamma (0)=x$ and $\gamma (1)=y$.The path must remain in $X$, including at its endpoints.