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

The constant maps form a connected subspace homeomorphic to $S$. Every $f\in C([0,1],S)$ is joined to the constant map $f(0)$ byUniform continuity of $f$ makes $s\mapsto H(s)$ continuous in the uniform metric. Thus the function space is a union of connected sets, each meeting the connected set of constant maps, and so