Given \(n\), show there is a continuous surjective map \(g: I \to I^n\). [Hint: Consider
\[ f \times f: I \times I \to I^2 \times I^2.] \]
Solution
Let \(p:I\to I^2\) be a continuous surjection, such as the Peano space-filling curve.
We first obtain surjections onto powers whose dimensions are powers of \(2\). Suppose \(q_k:I\to I^{2^k}\) is continuous and surjective. Then \[ q_k\times q_k:I^2\longrightarrow I^{2^{k+1}} \] is continuous and surjective, so \[ q_{k+1}=(q_k\times q_k)\circ p:I\longrightarrow I^{2^{k+1}} \] is continuous and surjective. Starting with \(q_0=\operatorname{id}_I\), induction gives such a map for every \(2^k\).
Given arbitrary \(n\ge1\), choose \(k\) with \(n\le2^k\) and compose \(q_k\) with the projection \[ I^{2^k}\longrightarrow I^n \] onto the first \(n\) coordinates. The resulting map \(I\to I^n\) is continuous and surjective.