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 $\gamma(1)=y$.
$$
\boxed{\forall x,y\in X\ \exists\gamma:[0,1]\to X\text{ continuous joining }x\text{ to }y.}
$$
The <path> must remain in $X$, including at its endpoints.

Solution

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

 Solution (source code)