Exercise 1, § 2

General Topology, VI

It is set in small type as supplementary.

Let $I$ be the closed cube of $\mathbf{R}^n$ which is the product of $n$ intervals identical with $[-\frac{1}{2}\pi, \frac{1}{2}\pi]$ . To each point $x = (x_i)$ of $I$ we make correspond the point $y = (y_j)$ of $\mathbf{R}^{n+1}$ such that

\begin{align*} y_1 &= \sin x_1 \\ y_2 &= \cos x_1 \sin x_2 \\ &\cdots \\ y_p &= \cos x_1 \cos x_2 \ldots \cos x_{p-1} \sin x_p \quad (2 \leq p \leq n-1) \\ &\cdots \\ y_n &= \cos x_1 \cos x_2 \ldots \cos x_{n-1} \sin 2x_n \\ y_{n+1} &= \cos x_1 \cos x_2 \ldots \cos x_{n-1} \cos 2x_n. \end{align*}

Show that the image of I under this mapping is the sphere $S_n$ , and that the restriction of this mapping to the interior of I is a homeomorphism onto a dense open set in $S_n$ . \*

Solution

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