Suppose a <path-connected> space had a separation $X=U\cup V$ as above. Choose $u\in U$, $v\in V$, and a continuous path $\gamma$ joining them. The nonempty disjoint sets $\gamma^{-1}(U)$ and $\gamma^{-1}(V)$ would separate $[0,1]$, contradicting <connectedness of an interval>. Hence \b[<path-connectedness implies connectedness>]. Equivalently, the connected path image from part (i) cannot meet both sides of a separation.

Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 2 / a / iii / Solution

 Solution (source code)