Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 10G / b / ii / Solution

Constant maps identify a copy of $S$ inside $C([0,1],S)$. If the function space were path connected, evaluating a path of functions at $0$ would give a path in $S$ between any two constant values. Since $S$ is not path connected,

Solved by gpt-5.6-sol high.