Since ${\mathbb C}^\times \mathbb{C}^\times \cong {\mathbb R}+\times \mathbb{R}_{+}\times S^1$ by polar coordinates, it suffices to show that $Hom({\mathbb R}+,{\mathbb R}+)$ \text{Hom}({\mathbb R}_+,{\mathbb R}_+)$ is uncountable. But for any real number $a$, $x\mapsto x^a$ gives an endomorphism of ${\mathbb R}+$. R}_+$. Explicitly, there are uncountably many group homomorphisms from ${\mathbb C}^\times$ to itself given by $re^{i\theta}\mapsto r^ae^{i\theta}$, for any real $a$.
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Since ${\mathbb C}^\times \cong {\mathbb R}+\times S^1$ by polar coordinates, it suffices to show that $Hom({\mathbb R}+,{\mathbb R}+)$ is uncountable. But for any real number $a$, $x\mapsto x^a$ gives an endomorphism of ${\mathbb R}+$. Explicitly, there are uncountably many group homomorphisms from ${\mathbb C}^\times$ to itself given by $re^{i\theta}\mapsto r^ae^{i\theta}$, for any real $a$. |
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