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The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue.

One possible proof uses the Brouwer fixed point theorem, and every proof I've seen following this form uses the fact that the "first quadrant" part of the sphere $S^{n-1}$ given by $\{(x_1,\dots,x_n)\in S^{n-1}:x_i\geq 0\}$ is homeomorphic to the closed ball $B^{n-1}$.

Is there a quick explanation of why these two are homeomorphic?

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    $\begingroup$ It has a projection by the origin to the regular simplex. If you move a regular simplex to the proper $\mathbb R^{n-1},$ centered at the origin, you can map the boundary to the circumscribed sphere, and scale the mapping of other points by the relevant factor, the origin stays put. $\endgroup$ – Will Jagy Oct 27 '13 at 3:48

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