Exercise 1893Topology › Topology of ℝⁿM1Added 3 Nov 2026

Ball and cube are homeomorphic

#homeomorphism#ball#cube

Statement

Show that the unit ball $B(0,1)\subseteq\mathbb{R}^n$ and the open cube $]-1,1[^n$ are homeomorphic.

Hint

Use a scaling by the sup-norm: $T(x)=x/\|x\|_\infty\cdot\|x\|_2$ (or directly $T(x)=\frac{\|x\|_2}{\|x\|_\infty}x$ when $x\neq0$ ).

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Map. Define $T(x)=\frac{\|x\|_2}{\|x\|_\infty}x$ for $x\neq0$ , $T(0)=0$ . This maps the cube $\|x\|_\infty<1$ onto the ball $\|x\|_2<1$ .

Inverse. The inverse is $T^{-1}(y)=\frac{\|y\|_\infty}{\|y\|_2}y$ , continuous.

Conclusion. Ball and cube are homeomorphic.