Let $\hat{\mathbb{C}}$ be the Riemann sphere.

Let $\overline{\mathbb{R}}$ be the extended real. (i.e. $\mathbb{R}\cup\{\infty,-\infty\}$)

Then, is $\hat{\mathbb{C}}\cong \overline{\mathbb{R}}\times\overline{\mathbb{R}}$?

If so, how do I prove it?

  • $\begingroup$ If you remove the extra point from $\hat{\mathbb C}$ you get something homeomorphic to the plane $\mathbb R^2$. Can you remove a point from $\overline{\mathbb R}\times \overline{\mathbb R}$ to get something homeomorphic to a plane? $\endgroup$ – Cheerful Parsnip Sep 17 '14 at 0:52

Try to show that $\overline{\Bbb R}$ is homeomorphic to the closed interval $[0,1]$. Can $\hat{\Bbb C}$ be homeomorphic to the square $[0,1]\times[0,1]$?


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